Groups and Topological Dynamics Volodymyr Nekrashevych

Groups and topological dynamics

Volodymyr Nekrashevych

E-mail address: [email protected] 2010 Mathematics Subject Classiﬁcation. Primary ... ; Secondary ... Key words and phrases. ....

The author ...

Abstract. ... Contents

Chapter 1. Dynamical systems 1 §1.1. Introduction by examples 1 §1.2. Subshifts 21 §1.3. Minimal Cantor systems 36 §1.4. Hyperbolic dynamics 74 §1.5. Holomorphic dynamics 103 Exercises 111

Chapter 2. Group actions 117 §2.1. Structure of orbits 117 §2.2. Localizable actions and Rubin’s theorem 136 §2.3. Automata 149 §2.4. Groups acting on rooted trees 163 Exercises 193

Chapter 3. Groupoids 201 §3.1. Basic deﬁnitions 201 §3.2. Actions and correspondences 214 §3.3. Fundamental groups 233 §3.4. Orbispaces and complexes of groups 238 §3.5. Compactly generated groupoids 243 §3.6. Hyperbolic groupoids 252 Exercises 252

iii iv Contents

Chapter 4. Iterated monodromy groups 255 §4.1. Iterated monodromy groups of self-coverings 255 §4.2. Self-similar groups 265 §4.3. General case 274 §4.4. Expanding maps and contracting groups 291 §4.5. Thurston maps and related structures 311 §4.6. Iterations of polynomials 333 §4.7. Functoriality 334 Exercises 340 Chapter 5. Groups from groupoids 345 §5.1. Full groups 345 §5.2. AF groupoids 364 §5.3. Homology of totally disconnected étalegroupoids 371 §5.4. Almost finite groupoids 378 §5.5. Bounded type 380 §5.6. Torsion groups 395 §5.7. Fragmentations of dihedral groups 401 §5.8. Purely infinite groupoids 423 Exercises 424 Chapter 6. Growth and amenability 427 §6.1. Growth of groups 427 §6.2. Groups of intermediate growth 427 §6.3. Inverted orbits and growth of wreath products 431 §6.4. Growth of fragmentations of D8 439 §6.5. Non-uniform exponential growth 447 §6.6. Amenability 448 Exercises 454 Bibliography 457 Chapter 1

Dynamical systems

A topological dynamical system H y X is an action of a semigroup H on a topolgical space X by continuous transformations. Classically, X is the phase space, the semigroup H represents time, and the action describes time evolution of the system. Accordingly, the acting semigroup is typically a subsemigroup of the additive group of real numbers (e.g., the semigroup of non-negative reals, the group of integers, or the semigroup of natural numbers). The subsequent chapters of the book will mostely deal with more “ex- otic” groups. But even in such cases, the groups often will be associated in a natural way to classical dynamical systems. The ﬁrst section of this chapter is a short overview of well known examples of dynamical systems. It introduces concepts that will be developed and generalized in the later parts of the book. The subsequent sections deal with more specialized topics in dynamical systems: subshifts, minimal homeomorphisms of Cantor sets, basic notions of hyperbolic dynamics, symbolic encoding of dynamical systems, and basic facts of holomorphic dynamics.

1.1. Introduction by examples

1.1.1. Irrational rotation. Consider the circle R{Z of real numbers modulo 1. It can be naturally identiﬁed with the complex unit circle T C by the map x ÞÑ e2πix. The circle R{Z is a group with respect to the addition. The above identiﬁcation of R{Z with the unit circle is an isomorphism of the additive group R{Z with the multiplicative group T .

1 2 1. Dynamical systems

Suppose that θ P R{Z is irrational. Consider the corresponding rotation

Rθ : x ÞÑ x θ. It is a homeomorphism of the circle, hence it generates an action of the infinite cyclic group Z by homeomorphisms. It is given by pn, xq ÞÑ nθ x for n P Z and x P R{Z. The central topic of topological dynamics is the study of topological properties of the orbits of a dynamical system H y X , i.e., the sets of the form Hx thx : h P Hu for x P X . For example, we may be interested in the cases when Hx is finite (or compact for a topological semigroup H), or in properties of the closure of Hx, etc.. If G is a group acting on a space X by homeomorphisms, then it naturally defines the orbit equivalence relation on X . It is given by x y ðñ Dg P G : gpxq y. The orbits of the action are the equivalence classes of this relation. It is natural to consider then the set GzX (or X {G for right actions) of orbits of the action. We introduce on it the smallest (coarsest) topology (i.e., topology with smallest set of open sets) for which the natural map X ÝÑ GzX is continuous. In other words, a subset A GzX is open if and only if its full preimage in X is open. The space GzX is frequently non-Hausdorff. For example, the following classical theorem of Kronecker [Kro84] implies that in the case of the irrational rotation Rθ the space of orbits of the action of Z on the circle has trivial topology (i.e., the only open sets are the empty set and the whole space).

Theorem 1.1.1. Every orbit tx nθ : n P Zu is dense in R{Z.

Proof. Denote, for a real number x, by fracpxq the fractional part of x, i.e., the unique number from the interval r0, 1q such that a ¡ fracpaq P Z. It is enough to prove that the orbit of 0 is dense in R{Z, since the orbit of an arbitrary point α P R{Z is obtained from the orbit of 0 by the rotation Rα. Consider an arbitrary positive integer N, and the arcs r0, 1{Nq, r1{N, 2{Nq, ... rpN ¡ 1q{N, 1q of the circle R{Z. Since the orbit tfracpnθq : n P Zu is inﬁnite, there exist integers n1 n2 such that fracpn1θq and fracpn2θq belong to the same arc. Then fracp|pn1 ¡ n2qθ|q 1{N. This proves that for every ¡ 0 there exists n P Z such that fracpnθq . Then the diﬀerence between consecutive entries of the sequence 0, fracpnθq, fracp2nθq,... is less than , hence for every α P R{Z there exists k such that fracp|α¡knθ|q . Consequently, the orbit of 0 is dense in R{Z. 1.1. Introduction by examples 3

Deﬁnition 1.1.2. An action H y X is minimal if every H-orbit Hx is dense in X .

It is easy to see that a group action G y X is minimal if and only if the space GzX is antidiscrete. Minimality is one of possible notions of irreducibility of topological dynamical systems, as the following lemma shows. (It also explains the origin of the term “minimal”, which referred to minimal invariant closed subsets.)

Lemma 1.1.3. A system H y X is minimal if and the only closed subsets Y X such that hpYq Y for every h P H are X and the empty set.

Proof. If H y X is not minimal, then there exists x P X such that the orbit Hx is not dense. Then its closure Y Hx satisﬁes hpYq Y for all h P H, and is not equal neither to X nor to H. In the other direction, if Y is H-invariant and closed, then for every x P Y the closure of the orbit Hx is contained in Y. So, if Y is non-emtpy and diﬀerent from X , then no H orbit of a point of Y is dense in X .

The circle is the quotient of R by the natural action of Z, and the rotation Rθ is lifted to the action on R of the transformation x ÞÑ x θ. The orbits of the irrational rotation are therefore equal to the images under the natural 2 quotient map R ÝÑ R{Z of the orbits of the action Z y R generated by the transformations x ÞÑ x 1 and x ÞÑ x θ. 2 2 Consider the natural action Z y R generated by the transformations a : px, yq ÞÑ px 1, yq, b : px, yq ÞÑ px, y 1q, 2 and consider the projection P : R ÝÑ R given by P px, yq x θy. We have P pap~vqq P p~vq 1 and P pbp~vqq P p~vq θ. In other words, the map 2 2 2 P projects the natural action Z y R to the action Z y R generated by x ÞÑ x 1 and x ÞÑ x θ. Denote by Q the composition of P with the natural quotient R ÝÑ R{Z. The orbits of the rotation Rθ ü R{Z are the Q-images of the sets of the form 2 2 Z v R . The segment r0, 1q R is a natural fundamental domain of the 2 quotient map R ÝÑ R{Z, and we can consider the part of Z v projected by P to r0, 1q. Every point of the Rθ-orbit of α P R{Z is represented then ¡1 2 ¡1 by exactly one point of the set P pr0, 1qq X pZ vq for v P Q pαq. See Figure 1.1, where an orbit of a point under a rotation is represented 2 2 in this way. The grid is the set Z v for some v P R . The transformation P is the projection onto the horizontal coordinate axis along lines parallel to the slanted lines shown on the picture. The strip between the slanted lines is the set P ¡1pr0, 1qq. The points of the grid inside the strip represent the points of the orbit of the rotation. The segments connecting neighboring 4 1. Dynamical systems

Figure 1.1. Irrational rotation

points of the orbit represent the action of Rθ. Let θ P r0, 1q (which we always can assume). Then for x P r0, 1q the point Rθpxq is represented either by x θ or by x θ ¡ 1 in r0, 1q. Therefore, on Figure 1.1, we move from the point representing x to the point representing Rθpxq either by adding p0, 1q or by adding p¡1, 1q. One can see these two cases as two types of edges of the broken line inside the strip on Figure 1.1: the vertical edges (corresponding to adding p0, 1q) and the diagonal edges (corresponding to adding p¡1, 1q). We can record the shape of the broken line by writing a two-sided inﬁnite sequence . . . dvvdvdvvdvdvvdvdvvd . . . , where d stands for “diagonal” and v for “vertical”. More formally, let xn be r q np q p q the point of 0, 1 representing Rθ x (i.e., xn frac x nθ ). Deﬁne the 1.1. Introduction by examples 5

Figure 1.2. Angle doubling

p q sequence an nPZ by the rule " v if xn 1 xn θ, an d if xn 1 xn θ ¡ 1.

We will study such sequences and their generalizations in subsequent sections of the book (see 1.2.5 and 3.5.2, for example). A multi-dimensional generalization of these sequences are quasi-crystals (see...) and, more generally, Cayley graphs of groupoids (see...).

1.1.2. Angle doubling map and one-sided shift. Consider the map f : x ÞÑ 2x pmod 1q on the circle R{Z. If we identify the circle R{Z with the complex unit circle by the map x ÞÑ e2πix, then f becomes z ÞÑ z2. The map f ü R{Z is a degree two covering. Every point x P R{Z has two preimages: x{2 and px 1q{2. Accordingly, as a dynamical system, we consider it to be an action of the semigroup of non-negative integers. In n 0 particular, the orbit of x is, by definition, the set tf pxqun0,1,2,.... Here f is the identity map. If x is a rational number, then f does not increase the denominator of x (it either does not change it or divides it by 2). Since we always can represent points of R{Z by points of r0, 1q, it follows that the f-orbits of rational points of R{Z are finite. Conversely, if x P R{Z has finite orbit, then there exist positive integers m n such that 2mx 2nx pmod 1q, which implies that x is rational. If x has a finite orbit under the action of a map f, then either x belongs to a cycle (is periodic) and f acts on the orbit of x as a cyclic permutation, or it is pre-periodic and the orbit contains two points of the form y{2 and 6 1. Dynamical systems py 1q{2 mapped by f to the same point. In the first case there exists a non- negative integer n such that f npxq x. The smallest such n is called the period or the length of the cycle. We have then that 2nx x pmod 1q, hence m x 2n¡1 , so that x is a fraction with an odd denominator. Conversely, if p P x q for p, q N, where q is odd, then the orbit of x is finite and can not be pre-periodic, since for every rational y one of the fractions y{2, py 1q{2 has an even denominator. It follows that x belongs to an f-cycle if and only if x is rational and has an odd denominator. In the pre-perodic case there exist smallest n ¡ 0 such that f npxq belongs to a cycle (i.e., has an odd denominator). We call this n the pre-period of p x. Then x is of the form 2nq , where p and q are odd. Let us represent the points of R{Z by their binary expansions. Namely, a point x P R{Z is represented by a sequence .a1a2a3 ... of zeros and ones so that 8 ¸ a x k pmod 1q. 2k k1 Denote by t0, 1uω the set of all infinite sequences of zeros and ones, and de- ω note by Φ : t0, 1u ÝÑ R{Z the natural map given by the binary numeration system: 8 ¸ a Φpa a ...q k pmod 1q. 1 2 2k k1 It is well known that the binary representation of real numbers is al- ω most one-to-one, i.e., Φ : t0, 1u ÝÑ R{Z is almost a bijection. The only ambiguity is

(1.1) .a1a2 . . . an10000 ... .a1a2 . . . an01111 ....

The space t0, 1uω comes with a natural direct product topology. It is deﬁned by the basis of open sets consisting of all sets of the form t P t uω u Ca1a2...an x1x2 ... 0, 1 : x1x2 . . . xn a1a2 . . . an , p q called sometimes cylindrical sets . Note that the sets Φ Ca1a2...an are closed m m 1 ¥ ¤ ¤ k ¡ intervals of the form 2k , 2k for k 0 and 0 m 2 1. ω ¡n A natural metric on t0, 1u is given by dpw1, w2q 2 , where n is the maximal length of a common beginning of w1 and w2. It is easy to see that we have |Φpw1q¡Φpw2q| ¤ dpw1, w2q, hence the map Φ is continuous. As we have seen above, Φ¡1pxq consists of one or two sequences, and all instances when Φ¡1pxq has two elements are described by (1.1). Namely, |Φ¡1pxq| 2 m if and only if x is of the form 2n for some natural numbers m and n. 1.1. Introduction by examples 7

If Φpa1a2a3 ...q x, then fpxq Φpa2a3a4 ...q. We get what is called a semiconjugacy implemented by Φ between two dynamical systems: f ü R{Z ω and the shift map s : a1a2a3 ... ÞÑ a2a3a4 ... on t0, 1u .

Deﬁnition 1.1.4. Let H y X and H y Y be two actions of the same semigroup on topological spaces. A semiconjugacy from the ﬁrst topological dynamical system to the second one is a continuous map Φ : X ÝÑ Y such that Φphxq hΦpxq for all h P H and x P X . If, additionally, Φ is a homeomorphism, then we say that it is a (topological) conjugacy.

In our case, the statement that Φ is a semiconjugacy is equivalent to the statement that the diagram s t0,1uω ÝÑ t0,1uω Φ Φ f R{Z ÝÑ R{Z commutes. The dynamical system s ü t0, 1uω is called the one-sided shift. This semiconjugacy can be used to prove many facts about the angle doubling map. For example, consider the following description of typical orbits of f ü R{Z.

Proposition 1.1.5. The set of points x P R{Z such that the orbit of x under f is dense is of full measure and co-meager in R{Z.

Recall that a set is called co-meager (or residual) if it is equal to intersection of a countable collection of open dense sets. By Bair Category Theorem ... such sets are non-empty for locally compact Hausdorﬀ spaces and for complete metric spaces. In these cases the notion of a co-meager set is a topological version of the notion of a set of full measure.

Proof. Consider the uniform Bernoulli measure µ on t0, 1uω. It is uniquely p q 1 determined by the condition µ Ca1a2...an 2n . Note that the length of p q r s 1 the subinterval Φ Ca1a2...an 0, 1 is also equal to 2n . It follows that ω ω Φ: t0, 1u ÝÑ R{Z maps the Bernoulli measure on t0, 1u to the Lebesgue measure on R{Z. In fact, Φ is an isomorphism of the corresponding measure spaces, since it is a bijection modulo a set of measure zero. Let w P t0, 1uω. If the orbit of w with respect to the shift action is dense ω in t0, 1u , then its Φ-image is dense in R{Z. The orbit of w is dense if and only if every ﬁnite word v P t0, 1u¦ appears as a subword in w. Equivalently, if the orbit is not dense, then there exists a ﬁnite word v P t0, 1u¦ that does ω ω not appear in w. Let Pv t0, 1u be the set of sequences w P t0, 1u not containing v as a subword. The set Pv is obviously closed. 8 1. Dynamical systems

If a ﬁnite word u contains v, then Cu X Pv H. For every n the number of words u P t0, 1un|v| not containing v as a subword is not more than p |v|¡ qn p2|v| ¡ 1qn. It follows that the measure of P is not more than 2 1 ¡ © v 2|v|n |v| n 2 ¡1 Ñ 0 as n Ñ 8. Consequently, P has measure zero. It follows 2|v| v that the set of points w with non-dense orbits has measure zero, as a countable union vPt0,1u¦ Pv of sets of measure zero. Consequently, the set of points w P t0, 1u¦ with dense orbit has measure one, which implies the same statement about R{Z. Note also that Pv has empty interior, since it has measure zero. Conse- quently, Pv is a closed nowhere dense set, and hence vPt0,1u¦ Pv is meager. The set ΦpPvq is also closed as a continuous image of a compact set. Sup- ¡1 pose that ΦpPvq has non-empty interior. Then Φ pΦpPvqq has non-empty interior. But for any subset A t0, 1uω, the set Φ¡1pΦpAqq is contained in the union of A with the countable set of sequences eventually equal to 000 ... or to 111 ... (since this is the set of points where Φ is not one-to-one). ¡1 It follows that Φ pΦpPvqq is contained in a union of a countable set with a nowhere dense closed set, hence it can not contain an open subset. Con- p q sequently, Φ Pv is closed and nowhere dense, and hence is meager, which p q implies that the set vPt0,1u¦ Φ Pv is meager.

A particular corollary of Proposition 1.1.5 is that there exists a point x P R{Z with a dense f-orbit. Existence of a dense orbit is another irreducibility notion for topological dynamical systems, weaker than minimality. We call it (in the case of group actions) topological transitivity. We will see later (Proposition 2.1.17) that a group action G y X on a second-countable completely metrizable space is topologically transitive if and only if every two non-empty open subsets of the space of orbits GzX have non-empty intersection. The angle doubling map f ü R{Z is very far from being minimal. Since the set of rational numbers with odd denominator is dense in R, we have the following property. Proposition 1.1.6. The union of all finite cycles under the angle doubling map is dense in R{Z. Density of the union of finite cycles is usually one of the ingredients of different definitions of “chaos”, see [LY75, Dev89, AH03].

1.1.3. Horseshoe and two-sided shift. Let S be a stadium-shaped re- 2 gion of R formed by a square and two half-discs shown on the left hand side part of Figure 1.3. Let Q be the square, and denote by D1,D2 the left and the right half-discs, as it is shown on the ﬁgure. 1.1. Introduction by examples 9

Figure 1.3. The horseshoe

Consider a continuous map F ü S mapping S to the horseshoe shaped region shown on the right hand side part of Figure 1.3. We assume that F acts on D1 as an affine map of the form x ÞÑ λx v1 for some λ P p0, 1{2q 2 and v1 P R , and on D2 as an affine map of the form x ÞÑ ¡λx v2 for the 2 same value of λ and for some v2 P R . We also assume that F pQq X Q is a disjoint union of two rectangles, and that the restriction of F onto the preimages of these rectangles are affine maps of the form px, yq ÞÑ pLx, λyq w1 and px, yq ÞÑ p¡Lx, ¡λyq w2 for 2 some w1, w2 P R . It is not very important how F acts on the rest of S. It is not hard to see, though, that we can choose F so that it is a diffeomorphism from S to F pSq.

Note that F ü D1 is a contraction, and has a unique fixed point p 1 np q Ñ Ñ 8 P 1¡λ v1 such that F x p as n for all x D1. We also have n F pD2q D1, hence F pxq Ñ p as n Ñ 8 for all x P D2. It follows that F npxq Ñ p for n Ñ 8 unless F npxq P Q for all n ¥ 0. Consider the set F ¡npQq of points x P S such that F npxq P Q. We define F ¡0pQq Q. The set F ¡1pQq is equal to the union of the two vertical rectangles V1 and V2 equal to the preimages of the two rectangles forming F pQq X Q, see the top part of Figure 1.4. The map F acts on V1 and V2 by affine transformations with the linear parts px, yq ÞÑ pLx, λyq and px, yq ÞÑ p¡Lx, ¡λyq, respectively. The set F ¡2pQq is equal to F ¡1pF ¡1pQq X F pQqq. The horseshoe F pQq intersects with the two rectangles forming F ¡1pQq in four rectangles (see the second row of Figure 1.4). Their preimages under F are four vertical rectangles (i.e., direct products of four segments in the horizontal side of Q with the full vertical side of Q). Continuing this way, we conclude that F ¡npQq is equal to a union of 2n vertical rectangles, and that each rectangle of F ¡npQq contains two rectangles of F ¡pn¡1qpQq. See, for example, the third row of Figure 1.4, where F ¡3pQq is described. 10 1. Dynamical systems

Figure 1.4. Sets F ¡npQq

If the side of Q has length 1, then the width of the rectangles forming ¡np q n ¡np q F Q is equal to λ . It follows that the set n¥0 F Q of points that stay in Q under positive iterations of F is a direct product of a horizontal Cantor set with the vertical side of the square Q. Its connected components are vertical subsegments of the square Q. We can label the rectangles of F ¡npQq by sequences of symbols T,B, where T and B stand for the top and the bottom rectangles of F pQq X Q, ¡n respectively. Namely, if x P F pQq, then the itinerary X1X2 ...Xn of x k is given by the condition that Xk T (resp., Xk B) if F pxq belongs to the top (resp., bottom) rectangle of F pQq X Q. Then the set of points with ¡n a given itinerary X1X2 ...Xn is one of the rectangles of F pQq. The two rectangles of F ¡pn 1qpQq contained in the rectangle of points with itineraries X1X2 ...Xn are the rectangles of points with the itineraries X1X2 ...XnT ¡np q and X1X2 ...XnB. It follows that n¥0 F Q is naturally homeomorphic ω to the direct product of tT,Bu with r0, 1s, where a set tX1X2 ...u¢r0, 1s for 1.1. Introduction by examples 11

Figure 1.5. Ranges of iterations

ω X1X2 ... P tT,Bu is identiﬁed with the vertical subsegment of Q consisting of all points x P Q such that for every n ¥ 1 the point F npxq belongs to the rectangle X of F pQq X Q. n ¡np q np q P The set n¥0 F Q is the set of points x such that F x Q for all n ¥ 0. Note, however, that the map F is not surjective on this set, i.e., there are points that do not have preimages under F . Let us describe the sets F npQq X Q. They are shown on Figure 1.5, for n 1, 2, 3, inside the left-hand side squares. The right-hand side of the ﬁgure shows their images under F . We obviously have F npxq X Q F pF n¡1pxq X Qq X Q. n n Similarly to the case of the sets Qn, we see that F pQq X Q consists of 2 horizontal rectangles of width L¡n. Each rectangle of F npQq X Q is labeled n ¡1 by itineraries X1X2 ...Xn P tT,Bu of its points under the map F . Each n rectangle X1X2 ...Xn of F pQq X Q contains two rectangles TX1X2 ...Xn n 1 and BX1X2 ...Xn of F pQq X Q. 12 1. Dynamical systems

np q X The intersection n¥1 F Q Q is the direct product of a vertical Cantor set with the horizontal side of the square. The Cantor set is naturally ¡ω identified with the space tT,Bu of left-infinite sequences ...X2X1 over the alphabet tT,Bu. ¡np q np q X Let us denote W n¥1 F Q , and W¡ n¥1 F Q Q. The set W is a Cantor set of vertical segments in the square Q. The set W¡ is a Cantor set of horizontal segments. We have F pW q W and F pW¡q n W¡. The set W is equal to the set of points x P S such that F pxq P Q for n all n ¥ 1, i.e., precisely to the set of points such that F pxq does not converge P np q P n kp q X to the fixed point p. Note that if x W , then F x k1 F Q Q, n which implies that the distance from F pxq to W¡ converges to zero as n n Ñ 8. Note also that F pxq P W for all n ¥ 1. Consequently, the n distance from F pxq to W X W¡ converges to zero. The map F acts as a homeomorphism on W W XW¡. n We see that either F pxq converges to the fixed point p (if x R W ) or the distance from F npxq to W goes to zero. It follows that W is an attractor.

Deﬁnition 1.1.7. Let f ü X be a continuous map, where X is compact. A compact set C X is called an attractor if there exists an open set U C np q such that C n¥1 f U .

Recall that the vertical lines forming W are labeled by right-infinite sequences X0X1 ... of the symbols tT,Bu, while the horizontal lines forming W¡ are labeled by the left-infinite sequences ...X¡2X¡1 over the same set of symbols. Every horizontal segment intersects every vertical segment in exactly one point of W ; and, conversely, every point of W is the intersection point of a horizontal and a vertical segment. We see that the points of W can be labeled by bi-infinite sequence ...X¡2X¡1X0X1 .... The sequence ...X¡2X¡1X0X1 ... is the itinerary of the corresponding point x in the sense n that F pxq belongs to the rectangle labeled by Xn for every n P Z. It is easy to see that this correspondence between the points and their itineraries is a homeomorphism. It also follows directly from the definition that this homeomorphism conjugates the action of F on W with the two-sided shift s given by

sp...X¡2X¡1X0X1 ...q ...Y¡2Y¡1Y0Y1 ..., where Yn Xn 1.

Note that the map F acts in a neighborhood of every point of W as an aﬃne transformation with linear part px, yq ÞÑ pLx, λyq or px, yq ÞÑ p¡Lx, ¡λyq, i.e., it locally expands the horizontal and contracts the vertical directions of the square Q. This is an example of a system belonging to the class of hyperbolic dynamical systems, which is the subject of Section 1.4. 1.1. Introduction by examples 13

Figure 1.6. Inverse limit of angle doubling maps

For a ﬁnite alphabet X, the two-sided full shift is the dynamical system Z pp q q Z y X , where the generator of Z acts by the homeomorphism s xn nPZ p q Z yn nPZ, where yn xn 1. Let us introduce on X the metric ¡k dppxnq, pynqq 2 , where k is the maximal non-negative integer such that x¡kxk 1 . . . xk¡1xk y¡ky¡k 1 . . . yk¡1yk. Note that this metric agrees with the direct product topology on XZ.

Let w pxnq be an arbitrary point of XZ. Denote by W pwq the set of points pynq of XZ such that . . . x¡2x¡1 . . . y¡2y¡1. The set W pwq is ω ω naturally homeomorphic to X , where the homeomorphism W pwq ÝÑ X erases the negative coordinates. Similarly, denote by W¡pwq the set of points pynq P XZ such that the non-negative coordinates of pynq and pxnq coincide. ¡ω The set W¡pwq is naturally homeomorphic to the space X of the left- inﬁnite sequences . . . y¡2y¡1. The space XZ is naturally homeomorphic to the direct product X¡ω ¢Xω, where the homeomorphism X¡ω ¢Xω ÝÑ XZ concatenates the corresponding sequences. Consequently, we have a natural decomposition of XZ into the direct product W¡pwq ¢ W pwq. n n Note that if w1, w2 P W pwq, then dps pw1q, s pw2qq Ñ 0 as n Ñ 8 for ¡n ¡n any metric on XZ. Similarly, if w1, w2 P W¡pwq, then dps pw1q, s pw2qq Ñ 0 as n Ñ 8.

1.1.4. Smale-Williams solenoid and the adding machine. Consider the backwards sequence

f f f f R{Z ÐÝ R{Z ÐÝ R{Z ÐÝ R{Z ÐÝ ¤ ¤ ¤ of the angle doubling maps f : x ÞÑ 2x pmod 1q from 1.1.2, see Figure 1.6. Let S be the inverse limit of this sequence. By definition, it is the set of sequences px1, x2,...q of points of R{Z such that fpxn 1q xn for all n. ω The topology is induced from the direct product topology on pR{Zq . Note 14 1. Dynamical systems that the map p fpx1, x2,...q pfpx1q, fpx2q,...q is a well defined homeomorphism of S, where the inverse homeomorphism is the shift px1, x2,...q ÞÑ px2, x3,...q. Definition 1.1.8. Let f ü X be a continuous map. Its natural extension is the homeomorphism fp induced by f on the inverse limit of the sequence f f f X ÐÝ X ÐÝ X ÐÝ ¤ ¤ ¤

The natural extension of the angle doubling map is called the Smale- Williams solenoid, see [Sma67] page 788. Note that S is a compact abelian topological group, since the map x ÞÑ 2x is an endomorphism of the compact abelian group R{Z, and S is the inverse limit with respect to these endomorphisms. Let us extend the symbolic representation of the angle doubling map described in 1.1.2 to the solenoid. Let ξ px1, x2,...q be a point of S. If .b1b2 ... is a binary representation of xn, then the binary representation of xn¡1 fpx1q is .b2b3 .... It follows that the coordinates of ξ are encoded by sequences of the form

p.a1a2 . . . , .a0a1a2 . . . , .a¡1a0a1a2 ...,...q. It is natural then to represent ξ by the bi-inﬁnite binary sequence

. . . a¡2a¡1a0.a1a2 ..., where .a¡n 2a¡n 3 ... is the binary sequence representing xn P R{Z. We have the same description of the identiﬁcation of the sequences representing the same point of the solenoid as in the case of the circle. Namely, p q p q two binary sequences an nPZ and bn nPZ represent the same point of S if and only if they are either equal to each other, equal to 000 ... and 111 ..., or are of the form . . . an¡1an10000 ... and . . . an¡1an01111 ... for some n P Z. One can show that the quotient of t0, 1uZ by this identiﬁcation rule is homeomorphic to S, and that the map on S induced by the two-sided shift s ü t0, 1uZ is topologically conjugate to the natural extension of the angle doubling map.

Let ξ be a point of S represented by a binary sequence . . . a¡2a¡1a0.a1a2 .... Let us call the sequences . . . a¡2a¡1a0 and .a1a2 ... the integral and fractional parts of ξ, respectively. The identiﬁcation rule for representation of points of S allows us to identify the fractional part with a point of the real interval r0, 1s via the natural map ¸8 ¡i .a1a2 ... ÞÑ ai2 . i0 1.1. Introduction by examples 15

Figure 1.7. The solenoid as a mapping torus

The integral part of ξ is naturally identified with a dyadic integer via ¸8 i . . . a¡2a¡1a0 a¡i2 P Z2. i0 Note that the integral and the fractional parts of ξ P S are uniquely defined, except for the points which can have fractional parts 0 or 1. Then a point can have fractional part 1 and integral part a P Z2, or fractional part 0 and integral part a 1 P Z2. It follows that S can be constructed by taking the direct product Z2 ¢ r0, 1s, and then identifying every point pa, 1q with pa 1, 0q. See Figure 1.7 for a schematical representation of this construction. In other words, the solenoid S is the mapping torus of the transformation a ÞÑ a 1 of the ring Z2 of dyadic integers. Definition 1.1.9. Let f ü X be a homeomorphism. Its mapping torus is the quotient of the space X ¢ r0, 1s by the identification px, 1q pfpxq, 0q.

The transformation τ : a ÞÑ a 1 of Z2 is called the adding machine or odometer. Its action on the binary sequences is given by the classical rule of adding one to a binary integer: " . . . a2a11 if a0 0; τp. . . a2a1a0q τp. . . a2a1q0 if a0 1. 16 1. Dynamical systems

Proposition 1.1.10. The odometer is a minimal homeomorphism of Z2.

See Deﬁnition 1.1.2 for the deﬁnition of a minimal action.

Proof. The group Z2 is the inverse limit (as a topological group) of the n n 1 n groups Z{2 Z with respect to the natural epimorphisms 1 2 Z ÞÑ 1 2 Z. The odometer is the inverse limit of the maps k ÞÑ k 1 acting on each n n Z{2 Z. These maps are transitive cycles on Z{2 Z, and every orbit of the action of the odometer on Z2 is mapped to these transitive cycles, which implies that every orbit is dense.

As a corollary we get the following topological property of the solenoid.

Proposition 1.1.11. Every path connected component of S is dense. In particular, S is connected but has uncountably many path connected components.

Note that fp multiplies the fractional parts by 2, which is a locally expanding map on r0, 1s. It is also multiplying the integer parts by 2, which is a contracting map on Z2. We see that, similarly for the two-sided shift, neighborhoods of points of the solenoid are decomposed into a direct product of an expanding and contracting directions of the dynamical system (i.e., is also a hyperbolic dynamical system, similarly to the two-sided shift). The solenoid can be also realized as an attractor of a diffeomorphism, see [Sma67, page 788] or [BS02, Section 1.9.]. Consider the region R inside 3 a torus in R obtained by rotating a disc D around a line in its plane not intersecting D. We call the images of D under the rotations meridional discs of R. Let F ü R be a map extendable to a diffeomorphism on a neighborhood of R such that F pRq is the region inside a torus winding twice around R, as it is shown on Figure 1.8. We assume that F maps every meridional disc of R to a smaller disc contained in a meridional disc of R. We also assume that F uniformly contracts the distances inside the meridional discs, and that it uniformly expands the distances in the direction perpendicular to the meridional discs. More explicitly, let us introduce a local coordinate system px, y, θq inside the torus, where x, y, for x2 y2 1 are the Cartesian coordinates in the disc D, and θ P p0, 2πq is the angle of rotation of the disc D around the axis of the torus.¨ We can define then F , for example, as F px, y, θq x cos t y sin t 4 2 , 4 2 , 2θ . One can show then that the intersection of the ranges of F n is homeomorphic to the solenoid, and that the restriction of F onto the intersection is topologically conjugate to the natural extension of the angle doubling map. 1.1. Introduction by examples 17

Figure 1.8. Solenoid map

Figure 1.9. Arnold’s cat

1.1.5. Anosov diﬀeomorphisms. Another classical example of a hyperbolic dynamical system is a hyperbolic automorphism of a torus, known in the literature as “Arnod’s cat map”. 18 1. Dynamical systems

Figure 1.10. Markov condition

2 2 Consider the map f ü ¢R {Z induced by the linear transformation of 2 1 2 with the matrix A . Since det A 1, the map f is a R 1 1 2 2 homeomorphism (and an automorphism of the group R {Z ). 2 The characteristic? polynomial of?A is λ ¡ 3λ 1 0, hence its eigen- values are λ 3 5 and λ¡1 3¡ 5 . Note that λ ¡ 1 and 0 λ¡1 1. 2 £ 2 £ ? 1 1¡ 5 ? 2 The eigenvectors of A are ~v1 ¡1 5 and ~v2 . Note that 2 1 they are orthogonal (A is symmetric). 2 An A-rectangle is a rectangle R R with the sides parallel to the 2 2 2 eigenvectors of A such that the quotient map R ÝÑ R {Z restricted to 2 2 the interior of R is injective. The images of A-rectangles in R {Z are also called A-rectangles. If R is an A-rectangle, and x P R, then we denote by Wspx, Rq the maximal segment inside R containing x and parallel to the contracting eigenspace (i.e., to the eigenspace of the eigenvalue λ¡1 1), and by Wupx, Rq we denote the maximal segment inside R containing x and parallel to the expanding eigenspace.

2 2 Deﬁnition 1.1.12. Let R be a ﬁnite set of A-rectangles R R {Z satisfying the following conditions: (1) The interiors of the rectangles R P R are disjoint, and union of their closures is the whole torus.

(2) If x belongs to the interior of R1 P R and fpxq belongs to the interior of R2 P R, then

ApWspx, R1qq WspApxq,R2q and

WupApxq,R2q ApWupx, R1qq. 2 2 Then the set R is called a Markov partition of A ü R {Z .

2 2 For example, we can construct a Markov partition of A ü R {Z in the following way. Consider the square formed by the lines parallel to the 1.1. Introduction by examples 19

Figure 1.11. Markov partition

eigenvectors with one vertex p0, 1q, and one side containing p0, 0q. Consider 2 2 all its translations by the elements of Z R . See the left-hand side of 2 Figure 1.11, where they are shown blue. The components of the part of R not covered by these squares are also squares. Their sides are parallel to 2 the eigenvectors of A and form one Z -orbit (red on Figure 1.11). Consider 2 the images of the two constructed Z -orbits of squares in the torus. We 2 2 get a partition of the torus R {Z into two squares with sides parallel to the eigenvectors of A. Their images under the action of A are shown on the right-hand half of Figure 1.11. It is easy to check that the constructed partition of the torus satisfies the conditions of Definition 1.1.12. 2 2 Let R be a Markov partition of A ü R {Z . Then for every R P R the intersections of the form ApRqXRi for Ri P R subdivide the rectangle ApRq into a finite number of disjoint sub-rectangles, by cutting the expanding direction into pieces. For each of these sub-rectangles R1 and x P R1 we have 1 Wspx, Rq Wspx, R q, see Figure 1.10. Note that an intersection ApRq X Ri can consist of several sub-rectangles of ApRq. Consider the oriented graph with the vertices identified with the elements of R and an edge from R to Ri for every rectangular piece of a non-empty intersection ApRq X Ri. Let us call this graph the structural graph of the Markov partition. For an edge e of the structural graph, denote by Re the corresponding (closed) piece of an intersection ApR1q X R2 for Ri P R (so that R1 and R2 correspond to the beginning and end of the edge e, respectively). For example, the structural graph of the Markov partition from Fig- ure 1.11 is shown on Figure 1.12. 20 1. Dynamical systems

Figure 1.12. Structural graph

Denote by M the set of all bi-inﬁnite paths in the structural graph. It is a subshift, i.e., a closed shift-invariant subset of the full shift Eω, where E is the set of edges of the graph.

Proposition 1.1.13. For every inﬁnite path w . . . e¡1e0e1 ... P M there p q P 2{ 2 np p qq P exists exactly one point φ w R Z such that A φ w Ren for all 2 2 n P Z. The map φ : M ÝÑ R {Z is a surjective semi-conjugacy from the 2 2 subshift s ü M to A ü R {Z .

Proof. Let . . . a¡1a0a1 ... P M. It follows from the deﬁnition of the map t φ that the image of the cylindrical set Ca0a1...an . . . e¡1e0e1 ... : ei u X ¡1p qX ¡2p qX X ai, i 0, 1, . . . , n is equal to the rectangle Ra0 A Ra1 A Ra2 ... ¡np q p q A Ran . It follows that φ Ca0a1...an is a rectangle R such that the side p q p q p q Ws x, R is equal to the side Ws x, Ra0 , and the side Wu x, R has length ¡n not more than Kλ , where K is the maximum length of the sides Wupx, Riq p q for all rectangles Ri of the Markov partition. It follows that φ Ca0a1...an , n ¥ 0, is a descending sequence of compact rectangles such that one side (parallel to the contracting direction) stays the same, while the length of the other side (parallel to the expanding direction) is exponentially decreasing.

Similarly, the set φpt. . . e¡1e0e1 ... : ei ai, i 0, ¡1,..., ¡nuq is a p q p q subrectangle R of Ra0 such Wu x, R is equal to the side Wu x, Ra0 , while the contracting sides Wspx, Rq form a nested sequence of subintervals of p q Ws x, Ra0 of exponentially decreasing lengths. It follows that for every ﬁnite path a¡na¡n 1 . . . an¡1an in the structural graph the set φpt. . . e¡1e0e1 ... : ei ai, ¡n ¤ i ¤ nuq is a rectangle with length both sides less than Kλ¡n for some ﬁxed K ¡ 0 and for λ ¡ 1. This implies that φ is continuous. It is easy to check that it is onto and a semi-conjugacy.

The encoding of points of the torus by bi-inﬁnite sequences is analogous to the encodings of the systems considered in 1.1.3 and 1.1.4. In fact, they are examples of a general construction, which will be studied in 1.4.11. 1.2. Subshifts 21

1.2. Subshifts

1.2.1. Definition and examples. Let H be a semigroup, and let X be a finite alphabet. We always assume that |X| ¥ 2. Consider the space XH with the topology of the direct product of discrete sets. In other terms, XH is the set of all maps f : H ÝÑ X, and the topology is defined by the basis of open cylindrical sets: t ÝÑ | u Cf0:AÝÑX f : H X : f A f0 , where A runs through the set of all finite subsets of H, and f0 runs through the set of all maps A ÝÑ X.

Note that Cf0:AÝÑX is also closed, since it is equal to the complement of A the union of the sets of the form Cf:AÝÑX for all f P X such that f f0. If H is the semigroup N of non-negative integers, we we represent the elements f P XN as sequences x0x1x2 ... fp0qfp1qfp2q .... Similarly, for H Z, we write the elements of XZ as bi-infinite sequences . . . x¡2x¡1 . x0x1 ..., where xn is the value of the element at n. We denote by a dot the place between the coordinates number -1 and 0. It follows directly from the definitions that for every element h P H the map f ÞÑ h ¤ f, where h ¤ f is defined by h ¤ fpxq fpxhq, is a (left) action of H on XH by continuous maps. We call the dynamical H system H y X the (full) H-shift. For example, the action of the generator 1 of N on XN is given in terms of sequences by

x0x1 ... ÞÑ x1x2 ..., while the action of the generator 1 of Z on XZ is given by

. . . x¡2x¡1 . x0x1 ... ÞÑ . . . x¡1x0 . x1x2 .... These maps are called the (full) one-sided and two-sided shifts, respectively.

Deﬁnition 1.2.1. An H-subshift is a dynamical system H y X , where X is a closed H-invariant subset of the full shift space XH . Here a subset Y X is said to be H-invariant if hY Y for every h P H.

One way to deﬁne a subshift is to take an arbitrary closed subset F X , and consider the intersection hPH hF. Another standard way is choose an element f of XH and take the closure of the H-orbit th ¤ f : h P Hu. 22 1. Dynamical systems

Deﬁnition 1.2.2. Let X XZ be a Z-subshift. Its language is the set WX p q P of all ﬁnite words v a0a1 . . . an such that there exists xn nPZ X such that xi ai for all i 0, 1, . . . , n. The following is straightforward.

Proposition 1.2.3. Let WX be the language of a subshift X XZ.A p q sequence xn nPZ belongs to X if and only if xnxn 1 . . . xm belongs to WX for all n ¤ m.

The following is a classical fact (see, for example [MH38, Theorem 7.2]). We say that a subshift is minimal if the Z-action on it is minimal, i.e., if all orbits of the shift are dense in it, see Deﬁnition 1.1.2. Note that in this context the word “minimal” has the usual sense: a subshift is minimal if and only if it does not contain a proper non-empty subshift. n Proposition 1.2.4. Let w P XZ. The closure of the orbit ts pwq : n P Zu of w in XZ is a minimal subshift if and only if for every ﬁnite subword v of w there exists R ¡ 0 such that every subword of w of length R contains v as a subword.

In other words, the closure of the orbit of w is a minimal subshift if and only if every ﬁnite subword of w appears in w inﬁnitely often with gaps of uniformly bounded length.

Proof. Let C be the closure of the orbit of w. Suppose that s ü C is minimal, and let v be a subword of w of length m. Then for every w1 P C n 1 there exists n P Z such that s pw q belongs to the cylindrical set Cv 1 t. . . x¡1x0x1 ... P C : x1x2 . . . xm vu, as the shift orbit of w is dense n in C. In other words, the sets of the form s pCvq for n P Z cover C. But n n n C is compact, so there exists a finite cover s 1 pCvq, s 2 pCvq,..., s k pCvq of n¡n C. It follows that for every n P Z there exists ni such that s i pwq P Cv. This exactly means that the word v appears in w on a uniformly bounded distance from any place in w. The converse statement (if every finite subword of w appears in w with uniformly bounded gaps, then the closure of the orbit of w is a minimal shift) is straightforward, and is left as an exercise. 1.2.2. Expansive actions. Definition 1.2.5. An action of a semigroup H on a metric space X is said to be expansive if there exists δ ¡ 0 such that if for all g P H we have dpgpxq, gpyqq δ then x y.

The deﬁnition of expansivity of an action does not depend on the metric if X is compact. Namely, we have the following equivalent deﬁnition. 1.2. Subshifts 23

Definition 1.2.6. Let X be a compact space. A neighborhood U X ¢ X of the diagonal tpx, xq : x P X u is called an expansion entourage for an action H y X if pgpxq, gpyqq P U for all g P H implies x y. Here U denotes the closure of U in X ¢ X . It is easy to check that an action H y X on a compact metric space X is expansive if and only if it has an expansion entourage. Let U be a closed expansion entourage for an action of H on a compact space X . For a finite subset A H, denote by UA the set of pairs px, yq P X ¢ X such that phpxq, hpyqq P U for all h P A. In other words, £ ¡1 UA h pUq, hPA where H acts on X 2 diagonally. Lemma 1.2.7. For every neighborhood of the diagonal W X ¢ X there exists a finite set A H such that UA W . Proof. It is enough to prove the lemma for the case when W is open. Then 2 2 X r W is compact, and for every px, yq P X r W there exists h P H such 2 ¡1 that phpxq, hpyqq R U. The set X r h pUq is open, so we have an open 2 2 ¡1 cover of X r W by sets of the form X r h pUq. There exists a finite 2 subcover, and its union is equal to X r UA for some finite A H. It follows that UA W . Expansive dynamical systems are chaotic in the sense that they exhibit sensitive dependence on the “initial conditions”. Namely, however small is the distance between the initial positions of two points x, y, as soon as x y there exists a moment g P H such that the distance between gpxq and gpyq is at least δ. This and similar conditions are ingredients of various definitions of chaos, see [LY75, Dev89, AH03]. Example 1.2.8. An action by isometries is obviously not expansive. In particular, a rotation of the circle and the odometer (see its definition before Proposition 1.1.10) are examples of non-expansive action. Example 1.2.9. The two-sided shift, the solenoid, and the Arnold’s Cat map are expansive. This follows from the corresponding local direct product decomposition into expanding and contracting directions. Namely, any two sufficiently close points x, y belong to one rectangle of such a decomposition. If x y, then either their projections onto the expanding side or the projections onto the contracting side of the rectangle are different. Then the distance between the corresponding projections of the points f npxq, f npyq will grow exponentially (if it is small) for positive or negative values of n, respectively. This proves that f npxq and f npyq can not be close to each 24 1. Dynamical systems

other for all n P Z. We will study this approach and this proof in more detail in 1.4.7.

The following description of expansive actions is a classical result, see....

Theorem 1.2.10. An action H y X of a semigroup H on a compact totally disconnected space X is expansive if and only if the dynamical system H y X is topologically conjugate to an H-subshift. Proof. Let δ be as in Definition 1.2.5. Consider a finite partition U of X into clopen sets of diameters less than δ. For a point x P X , define its itinerary as the map Ix : H ÞÑ U given by the rule

hpxq P Ixphq. U The map x ÞÑ Ix is a continuous map from X to H , since it is locally constant on each coordinate. It follows that its range is a compact subset U of H . We have ghpxq P Ixpghq and ghpxq P Ihpxqpgq. It follows that h ¤ Ixpgq Ixpghq Ihpxqpgq, so that h ¤ Ix Ihpxq, i.e., that the map U I¤ : X ÝÑ H is H-equivariant.

The map x ÞÑ Ix is also injective, since Ix Iy implies that the distance from hpxq to hpyq is less than δ for every h P H. It follows that x ÞÑ Ix is a homeomorphism from X to its images, since every injective continuous map from a compact space to a Hausdorff space is a homeomorphism onto the image. t u 1.2.3. Shifts of finite type. Let P Cfi:AiÝÑX iPI be a collection of cylindrical subsets of XH . Consider now the set of all sequences w P XH such that configurations fi : Ai ÝÑ X do not appear in any shifts of w. P H ¤ | Namely, consider the set XP of elements w X such that h w Ai fi for all i P I. We say that XP is the shift defined by the set of prohibited configurations P. Note that XP is closed and H-invariant, i.e., that it is a subshift. Every subshift can be defined by some collection of prohibited configurations. Definition 1.2.11. A subshift X XH is a shift of finite type if there exists a finite set of prohibited configurations defining X .

¦ In particular, a Z or N-shift X is defined by a finite collection A X of finite prohibited words. A sequence w belongs to the corresponding subshift if and only if no subword of w belongs to A. Note that a subset of XH is a union of a finite number of cylindrical sets if and only if it is clopen. It follows that a subshift X XH is of H finite type if and only if there exists a clopen set U such that X X r ¡1p q ¡1p q hPH h U , where h U denotes the full preimage of U under the action 1.2. Subshifts 25

H of h. Replacing U by V X r U, we get another deﬁnition of shifts of ﬁnite type.

H Lemma 1.2.12. A subshift X X is of ﬁnite type if and only if there ¡1p q exists a clopen set U such that X hPH h U . More generally, we adopt the following deﬁnition.

H Definition 1.2.13. Let X X be a subfshift. A subshift X1 X is of relative finite type in X if there exists a clopen subset U X such that ¡1p q X1 hPH h U . Let G be a group, and let X be a finite alphabet. For a finite subset B G B G B G, consider the alphabet X and the map βB : X ÝÑ pX q given by the equality βBpfqpgq pg ¤ fq|B.

Proposition 1.2.14. The map βB is a G-equivariant homeomorphic embedding.

Proof. We have

βBph ¤ fqpgq ppghq ¤ fq|B βBpfqpghq h ¤ βBpfqpgq, hence βB is G-equivariant. P p qp ¡1 q Choose an element g0 B. Then the value of βB f g0 g as a function G B ÝÑ G on the point g0 is equal to fpgq. It follows that f P X can be reconstructed from βBpfq, i.e., that βB is injective. It is obviously continu- G B G ous and the spaces X and pX q are compact and Hausdorﬀ, hence βB is a homeomorphic embedding. G B G Deﬁnition 1.2.15. The map βB : X ÝÑ pX q is called the block map (or sliding block code) with the window B.

For example, in the case G Z, it is natural to consider a window of the form t0, 1, . . . , n ¡ 1u. Then XB is the set of words of length n, and the block map transforms a sequence . . . x¡1.x0x1 ... over the alphabet X to the sequence

... px¡1x0 . . . xn¡2q.px0x1 . . . xn¡1qpx1x2 . . . xnq ... over the alphabet Xn. It is easy to see that the block map is a shift- equivariant homeomorphic embedding. Let G be a group generated by a ﬁnite set S. Choose for every gener- 2 ator s P S a set As X of pairs of letters, and deﬁne the corresponding topological Markov shift as the set of all G-sequences f P XG such that

pfpgq, fpsgqq P As 26 1. Dynamical systems

Figure 1.13. Fibonacci s.f.t. for every g P G and s P S. It is easy to see that this is a shift of ﬁnite type. For example, if G Z, then it is natural to consider S t1u, and then the corresponding topological Markov shifts are given by a set A X2 of p q allowed transitions. A sequence xn nPZ belongs to the corresponding shift if and only if xnxn 1 P A for all n P Z. One can represent the set A by an |A| ¢ |A| transition matrix. Its entries ax,y are indexed by the letters x, y of X, and we have " 1 if xy P A, a x,y 0 otherwise.

Example 1.2.16. Consider the subshift of t0, 1uZ deﬁned by the set of allowed transitions t00, 01u. In other words, a sequence belongs to F if and¢ only if it does not contain 11 as a subword. Then transition matrix is 1 1 . 1 0

Another way of representing topological Markov shifts is to use graphs with the set of vertices X, where we have an arrow from x to y if and only if xy P A. Then the subshift is the set of all bi-infinite sequences that can be read along bi-infinite paths in the graph. Example 1.2.17. The subshift from Example 1.2.16 is described by the graph shown on Figure 1.13. Example 1.2.18. Consider the graph shown on Figure 1.12. The corresponding topological Markov chain describes the subshift encoding the Arnold’s Cat map, see Proposition 1.1.13. Proposition 1.2.19. Let G be a group with a finite generating set S S¡1. For every shift of finite type X XG there exists a block map conjugating X with a topological Markov shift.

t ÝÑ um Proof. Let fi : Ai X i1 be a finite set of prohibited configurations m p Y t uqR p Y defining X . Let R be such that i1 Ai S 1 . Denote B S t1uqR. We can define X by a set of prohibited configurations f : B ÝÑ X defined on B. Equivalently, we can find a finite subset Y XB of maps 1.2. Subshifts 27

B ÞÑ X such that f P XG belongs to X if and only if for every g P G the restriction of g ¤ f to B belongs to Y. G Consider the block map βB with the window B. For an element f P Y and g P G we will denote by fg the function fg : gB ÝÑ X given by ¡1 fgphq fpgqpg hq (check...), where fpgq : B ÝÑ X is the element of Y corresponding to g P G (i.e., the value of f P YG at g). G An element f P Y belongs to βBpX q if and only if for every pair g1, g2 P p q p q P X P G G we have fg1 h fg2 h for every h g1B g2B. In other words, f Y p q belongs to βB X if and only if the functions fg1 and fg2 agree on the intersection of their domains. Consider the topological Markov S shift consisting of all sequences f P G Y given by the condition that fg and fsg agree on the intersection gBXsgB of their domains, where g P G and s P S. Let us show that S coincides with βBpX q.

The inclusion βBpX q S is obvious. Suppose that f P S. We have to P p q p q prove that for any g1, g2, h G we have fg1 h fg2 h whenever both sides of the equality are defined. By G-equivariance, it is enough to prove this statement for the case h 1. We have 1 P g1B X g2B if and only if g1 and g2 are on distance at most R from 1. Consequently, it is enough to prove p q p q P p q p q that fg1 1 fg2 1 for every g1, g2 B, i.e., that fg 1 f1 1 for every g P B. Write g as a product of at most R generators g s1s2 ¤ ¤ ¤ sn. Then, by definition of S, we have p q p q p q p q f1 1 fsn 1 fsn¡1sn 1 ... fs1s2¤¤¤sn 1 , which finishes the proof. 1.2.4. Substitutional subshifts. Let X be a finite alphabet. Denote by ¦ X the free monoid generated by X, i.e., the set of all finite words x1x2 . . . xn over the alphabet X, including the empty word ∅. It is a semigroup (monoid) with respect to the operation of concatenation. We say that v is a subword ¦ ¦ of w P X if there exist v1, v2 P X such that w v1vv2. If x P X is a letter, then we sometimes say that x appears in w if x is a subword of w. The notion of a sub-word of an infinite word (sequence) is defined analogously. A substitution is an endomorphism σ : X¦ ÝÑ X¦ of the monoid. It is defined by the values σpxq P X¦ on the elements of X. Namely, for every word x1x2 . . . xn we have σpx1x2 . . . xnq σpx1qσpx2q . . . σpxnq. ¦ ¦ If w . . . x¡2x¡1 . x0x1 ... is an element of XZ, and σ : X ÝÑ X is a substitution, then we denote by σpwq the infinite word

. . . σpx¡2qσpx¡1q . σpx0qσpx1q ..., where dot, as before, denotes the place between the coordinates number ¡1 and 0. 28 1. Dynamical systems

Definition 1.2.20. The substitutional subshift generated by σ : X¦ ÝÑ X¦ is the set of all bi-infinite words w P XZ such that for every finite subword v n of w there exists x P X and n P N such that v is a subword of σ pxq. In other words, the language of the subshift generated by σ is the set of all subwords of words of the form σnpxq for x P X. Note that in order for the substitutional subshift to exist, the length of the word φnpxq must go to infinity for some letter x P X. Another (probably more common in the literature) definition of the substitutional subshift is to choose a particular x P X such that the length of σnpxq goes to infinity and to consider the set of all bi-infinite sequences w such that every finite subword of w is a subword of σnpxq for some n. Note that if F is the substitutional subshift generated by σ : X¦ ÝÑ X¦, then F is σ-invariant, i.e., σpFq F. The range σpFq is usually not equal p q P to F, but if m is the maximum of the lengths of the words σ x , x X, then m¡1 kp p qq k0 s σ F F, where s is the shift. Example 1.2.21. Let X t0, 1u, and define an endomorphism σ : X¦ ÝÑ X¦ by σp0q 01, σp1q 10. The corresponding substitutional shift is called the Thue-Morse subshift. The first letter of σp0q is 0. It follows by induction that the word σn 1p0q begins with σnp0q for every n. Therefore, the sequence σnp0q naturally converges to one right-infinite sequence 01101001100101101001011001101001 .... This particular sequence is called the Thue-Morse sequence. It is easy to see that an infinite sequence belongs to the Thue-Morse subshift if and only if all its finite subwords are subwords of the Thue-Morse sequence. Thue-Morse sequence was defined independently by E. Prohuet [Pro51], A. Thue [Thu12], and M. Morse [Mor21]. E. Prouhet implicitly discov- ered the sequence as a solution of a particular case of Prouhet-Tarry-Escott problem, see Exercise 1.16. A. Thue considered it as an example of an infinite cube-free sequence, see Exercise 1.16. M. Morse gave it, essentially as an example of a minimal infinite subshift. See ... [literature] for more properties of the Thue-Morse sequence Example 1.2.22. Consider now the substitution σp0q 01, σp1q 0. Here too the word σnp0q is a beginning of σn 1p0q, and in the limit we get an infinite sequence 0100101001001010010100100101001001 ... 1.2. Subshifts 29 called the Fibonacci word. The corresponding subshift consisting of all bi- infinite sequences whose finite subwords are subwords of the Fibonacci word is called the Fibonacci substitutional subshift (not to be confused with the Fibonacci shift of finite type from Example 1.2.16).

Note that in general the set of ﬁnite subwords of the subshift Fσ generated by a substitution σ : X¦ ÝÑ X¦ may be strictly smaller that the set of all ﬁnite subwords of the words of the form σnpxq for x P X. For example, ¦ ¦ if σ : t0, 1u ÝÑ t0, 1u is given by σp0q 10, σp1q 1, then Fσ consists of one sequence ... 111 ....

Proposition 1.2.23. Let Fσ be the subshift generated by the a substitution σ : X ÝÑ X¦. Then the following conditions are equivalent.

(1) For every x P X there exists w P Fσ such that x appears in w.

(2) The set of ﬁnite subwords of elements of Fσ is the same as the set of subwords of the words of the form σnpxq for x P X and n ¥ 0. (3) For every x P X there exists n ¥ 1 and y P X such that σnpyq ¦ v1xv2 for some non-empty words v1, v2 P X . Proof. Let us show that (1) implies (2). Let v be a subword of σnpxq. n n There exists w P Fσ containing x. Then σ pwq P Fσ and it contains σ pxq, hence it contains v. We obviously have that (2) implies (1), since every letter x is equal to σ0pxq, by deﬁnition.

Let us show that (3) implies (1). Let x1 x P X be an arbitrary letter. Then, there exists n1 and x2 P X such that x1 is appears strictly n1 inside σ px2q. Inductively, there exists xk 1 P X and nk ¥ 1 such that xk n appears strictly inside σ k pxk 1q. Choosing a convergent subsequence of the n1 n2 ¤¤¤ n sequence of the words σ k pxk 1q we can find a bi-infinite sequence w P Fσ containing x. Conversely, if (1) is satisfied, then for every x P X there exist letters a, b P X such that axb is a subword of some sequence w P Fσ. Then, by definition of Fσ there exists y P X and n ¥ 1 such that axb is a subword of n σ pyq. ¦ ¦ Proposition 1.2.24. Let σ : X ÝÑ X be a substitution, and let pFσ, sq be the subshift generated by it. Then the following two conditions are equivalent. (1) There exists N ¥ 1 such that for every pair of letters x, y P X the letter y appears in σN pxq.

(2) The dynamical system pFσ, sq is minimal, for every letter x P X n there exists w P Fσ containing x, and the length of σ pxq goes to inﬁnity for every x P X. 30 1. Dynamical systems

Substitutions satisfying condition (1) of Proposition 1.2.24 are called primitive. references...

Proof. Let us prove the implication (1)ùñ(2). Let N be as in (1). Then for every x P X the word σN pxq contains all letters of the alphabet X, hence the length of σN pxq is at least |X| ¡ 1. It follows that the length of σN k pxq is at least |X|k for every k ¥ 1. Consequently, the length of σkpxq goes to infinity. It is also easy to see that if (1) is satisfied, then the condition (3) of Proposition 1.2.23 is satisfied, hence every letter of X appears in some element of Fσ.

It remains to show that Fσ is minimal. Suppose that L is larger than N the length of every word of the form σ pxq for x P X, and let w P Fσ be N arbitrary. A shift of w belongs to σ pFσq. It follows that every subword of w of length L contains a subword of the form σN pxq for some x P X. Consequently, every subword of w of length L contains every letter of X. It n follows that for all n ¥ 1, x P X, w P Fσ the word σ pxq is a subword of w, which implies that the subshift Fσ is minimal by Proposition 1.2.4. Let us show that (2) implies (1). Assume that (2) is satisﬁed. Then for every sequence w P Fσ, every x P X, and every k P Z there exists n P Z such that |k¡n| ¤ M and wpnq x. In other words, every letter appears in every word w P Fσ with bounded gaps. By the conditions of the proposition, there exists N such that the lengths of σN pxq are bigger than M for all x P X. N Then σ pxq will contain every letter y P X for every x P X.

If Fσ is minimal, but a letter x P X does not appear in a sequence w P Fσ, then the letter x does not appear in any sequence w P Fσ. It follows that there exists n such that x does not appear in any word σnpyq, y P X. Then ¦ Fσ is generated by the restriction of σ to the monoid pX r txuq . In other words, the condition that every letter appears in some element of the subshift is not restrictive if we want to describe all minimal substitutional subshifts. Here are some examples of minimial substitutional shifts not satisfying the other condition of the Proposition 1.2.24 (that the length of σnpxq always goes to infinity). Example 1.2.25. Let X t0, 1, 2u, and consider the substitution σp0q 021, σp1q 10, σp2q 2. Then the length of σnpxq goes to infinity if and only if x 2. It is easy to see that the subshift generated by σ is the set of all bi-infinite sequences obtained from the sequence belonging to the Thue-Morse subshift by inserting a 2 after every 0. 1.2. Subshifts 31

Example 1.2.26. Consider the Chacon substitution σp0q 0010, σp1q 1. n Denote bn σ p0q. In particular, b0 0 and b1 0010 b0b01b0. It follows by induction that bn 1 σpbnq σpbn¡1bn¡11bn¡1q bnbn1bn. The recurrent formula bn 1 bnbn1bn implies that the subshift generated by the Chacon substitution is minimal.

The last example is actually conjugate to a subshift generated by a primitive substitution. Namely, consider the substitution ϕp0q 0012, ϕp1q 12, ϕp2q 012. n It is primitive, so it generates a minimal subshift. Define vn ϕ p0q, and 1 let vn be the word vn in which the first symbol 0 is replaced by 2. 1 Let us show by induction that vn 1 vnvn1vn. It is true for n 0. Suppose that it is true for n, let us show it for n 1. We have vn 2 p q p q p 1 q p 1 q p q ϕ vn ϕ vn 12ϕ vn . The word ϕ vn is obtained from ϕ vn by replacing p 1 q p q the initial 0012 by 012. It follows that 2ϕ vn is obtained from ϕ vn by replacing the initial 0012 by 2012, i.e., it is obtained from ϕpvnq by replacing p 1 q 1 1 the initial 0 by 2. Consequently, 2ϕ vn vn 1, and vn 2 vn 1vn 11vn 1. We see that if σ is the Chacon substitution from Example 1.2.26, then σnp0q is obtained from ϕnp0q by replacing all symbols 2 by 0. In the other direction, it is easy to prove by induction that ϕnp0q is obtained from σnp0q by replacing every subword 10 of σnp0q by 12. It follows that the subshifts generated by σ and ϕ are topologically conjugate. For more on the Chacon substitution, see [Fer02]. On the other hand, up to topological conjugacy, the class of minimal substitutional subshifts coincides with the class of subshifts generated by primitive substitutions. Proposition 1.2.27. Let X XZ be a substitutional subshift. If it is minimal, then there exists a finite alphabet Y and a primitive substitution ¦ φ : Y ÝÑ Y such that the subshift Fφ generated by φ and the subshift X are topologically conjugate.

Proof. We may assume that every letter x P X belongs to the language of X , i.e., appears in some element of X . Otherwise, we can remove all such letters from the alphabet, and replace σ by an iterate, so that the removed letters do not appear in the values of σ on the remaining letters. Since X is minimal, every letter x P X appears in every element of X with uniformly bounded gaps between consecutive appearances, see Propo- n sition 1.2.4. Let x0 P X be such that the length of σ px0q goes to inﬁnity. Then there exists N ¡ 0 such that every word of length N in the language 32 1. Dynamical systems

WX of X contains x0. It follows that for every word v P WX the length of n σ pvq goes to inﬁnity. Let Y be the set of elements of WX of length N. For ¦ ¦ every x1x2 . . . xN P Y X , compute σpx1x2 . . . xN q a1a2 . . . am P X , and let σpx1q a1a2 . . . ak. Consider the word ¦ φpvq pa1a2 . . . aN qpa2a3 . . . aN 1q ... pakak 1 . . . ak N¡1q P Y .

Note that k n ¡ 1 ¤ N, since a1a2 . . . am a1a2 . . . akσpx2x3 . . . xN q. It is good to imagine x1x2 . . . xn P Y as a copy of the letter x1 P X decorated by the “future” word x2 . . . xn. Then φ becomes the substitution induced by σ on the decorated version of the alphabet. It is easy to prove by induction that φnpvq for v P Y is defined using σn by the same rule as φ was defined using σ. It follows that Fφ is obtained from X by conjugating by the block map with window on width N. Note that the length of φnpyq goes to infinity for every y P Y, and every letter y P Y is contained in every element of Fφ. Proposition 1.2.24 shows that φ is primitive. 1.2.5. Complexity and entropy.

Deﬁnition 1.2.28. Let F be a subshift, and let WF be its language. The complexity function of F is

pF pnq |tv P WF : |v| nu|, where |v| denotes the length of v.

The complexity functions is sometimes called the factor complexity, block growth, or subword complexity. It was introduced by G. Hedlund and M. Morse in [MH38]. See [CN10] for an overview of the main results on this subject.

Proposition 1.2.29. The function pF pnq is non-decreasing. Moreover, if pF pnq pF pn 1q for some n, then pF pnq is bounded, and F is ﬁnite.

We have pF pn mq ¤ pF pnqpF pmq for all n, m ¥ 1. The limit of the log pF pnq sequence n exists and is equal to its inﬁmum.

Proof. Every word v P WF of length n can be continued, by adding one 1 letter to its end, to a word v P WF of length n 1. If v1 v2, then their 1 1 ÞÑ 1 continuations v1, v2 are also diﬀerent. It follows that v v (for any choices of the continuations v1) is a one-to-one map from the set of words of length n to the set of words of length n 1 of the language WF . If pF pnq pF pn 1q, then this map is a bijection. It follows that every word x1x2 . . . xn P WF has a unique continuation x1x2 . . . xnxn 1 to a word of length n 1 belonging to WF . Then the word x2x3 . . . xn 1 also has a unique continuation, which implies that every word of length n has a unique continuation to a word of length n 2. By induction, we get that for every m ¡ n every word 1.2. Subshifts 33

v P WF is a preﬁx of a unique word w P WF of length m. Consequently, pF pmq pF pnq for every m ¡ n, and pF is bounded.

Every word v P WF of length n m can be split into a preﬁx of length n and a suﬃx of length m, so pF pn mq ¤ pF pnqpF pmq. The statement about the limit follows from the classical Fekete’s lemma [Fek23] and [PS72, p q Problem 98] applied to log pF n .

Proposition 1.2.30. Let F1 and F2 be topologically conjugate subshifts. Then there exists c ¡ 1 such that we have p ¡ q ¤ p q ¤ p q pF1 n c pF2 n pF1 n c for all n ¡ c.

Z P 2n 1 Proof. Let Xi be the alphabet such that Fi Xi . Suppose that v Xi is a word of odd length 2n 1, and denote by Cv,i the set all sequences p q P ÝÑ xn nPZ Fi such that x¡nx¡n 1 . . . xn¡1xn v. Let Φ : F1 F2 be p q P a topological conjugacy. Then for every sequence w xn nPZ Fi the t u intersection n¥0 Cx¡nx¡n 1...xn¡1xn,i is equal to w . Consequently, there p q P exists k such that Φ Cx¡kx¡k 1...xk¡1xk,1 Cy0,2 for some letter y0 X2. It follows from compactness of F1 that there exists K ¥ 0 such that for every P 2K 1 P p q word v X1 there exists yv X2 such that Φ Cv,1 Cyv,2. Then we have that Φ is given by the “centered” block map: pp q qp q Φ xn nPZ n yxn¡K xn¡K 1...xn K¡1xn K . For every n ¡ 2K 1, the constructed block map deﬁnes a surjective map from the set of words of length n of the language of F1 to the set of words of length n ¡ 2K of the language of F2: ÞÑ x1x2 . . . xn yx1x2...x2K 1 yx2x3...x2K 2 . . . yxn¡2K xn¡2K 1...xn . p ¡ q ¤ It follows that pF2 n 2K pF1pnq.

log pF pnq As a corollary of Proposition 1.2.30 we get that the limit limnÑ8 n (which exists by Proposition 1.2.29) depends only on the topological conjugacy class of the subshift. It is called the entropy of the subshift.

Example 1.2.31. Let F XZ be a topological Markov shift with the transition matrix A. Then, for all n ¥ 2, the complexity pF pnq is equal to the sum of the entries of the matrix An¡1. The entropy of F is equal therefore to the logarithm of the spectral radius of A. Example 1.2.32. If F is the Fibonacci shift of ﬁnite type from Exam- ple 1.2.16, then pF pnq is the Fibonacci sequence 2, 3, 5, 8,.... It follows that the entropy is equal to the logarithm of the golden mean. 34 1. Dynamical systems

Entropy is an important invariant of dynamical systems and a classical subject originating from the work of Shannon on information theory. The notion deﬁned above is a particular case of topological entropy .... more on history ... literature... The following theorem was proved in [Gri73]. See a short proof in [CN10, Theorem 4.4.4]. Theorem 1.2.33. For every h ¡ 0 and every integer k such that h log k there exists a minimal subshift F t1, 2, . . . , kuZ of entropy h.

Substitutional dynamical systems, on the other hand, are examples of subshifts of zero entropy. Theorem 1.2.34. Let F be a minimal substitutional subshift. Then there ¡ p q ¤ ¥ exists C 1 such that pFσ n Cn for all n 1. Proof. It follows from Propositions 1.2.27 and 1.2.30 that we may assume ¦ that F Fσ is generated by a primitive substitution σ : X ÝÑ X . Moreover, we may assume that for every x P X the word σpxq contains all letters of the alphabet (otherwise, replace σ by a high enough iterate). Then for every x, y P X and k ¥ 1 we have |σkpxq| ¤ |σk 1pyq|, since σkpxq is a subword of σk 1pyq. k 1 k Let M maxxPX |σpxq|. We have then |σ pxq| ¤ M|σ pxq| for all x P X and k ¥ 1. It follows that |σkpxq| ¤ M|σkpyq| for all k ¥ 1 and x, y P X. Let n ¥ 1, and let k be the smallest integer such that n ¤ |σkpxq| for all k¡1 x P X. Then there exists x1 P X such that |σ px1q| n. Then, for every x P X, we have k k¡1 2 k¡1 2 |σ pxq| ¤ M|σ pxq| ¤ M |σ px1q| M n. P p q P Suppose that v WFσ has length n. Let xn nPZ Fσ be such that v k k k k k is a subword of . . . σ px¡2qσ px¡1q.σ px0qσ px1q .... Since all words σ pxq are of length at least n, there exists a word xixi 1 of length 2 such that v k k is a subword of σ pxixi 1q. The length of σ pxixi 1q is strictly less than 2 k 2 2M n, hence σ pxixi 1q has not more than 2M n ¡ n subwords of length p q ¤ p qp 2 ¡ q n. It follows that pFσ n pFσ 2 2M 1 n, so the Theorem is valid for p qp 2 ¡ q C pFσ 2 2M 1 . See a similar proof of Theorem 1.2.34 (for the case of a primitive substitution) using Perron-Frobenius theorem in [Que87, Proposition V.19]. 1.2. Subshifts 35

Minimal systems of various subexponential complexity have been constructed in the literature. For example, the following is a result of J. Goyon, see [Fer99, p. 149].

Theorem 1.2.35. Let φ : N ÝÑ N be such that there exist r, h ¡ 1 such that for all k ¥ 1 we have φprk 1q ¤ hφprkq. Then there exists a minimal subshift F and positive constants C1,C2 such that

C1nφpnq ¤ pF pnq ¤ C2nφpnq for all n ¥ 1.

For example, there exist minimal subshifts whose complexities are of α α α α the form n 0 plog nq 1 plog log nq 2 ¤ ¤ ¤ plog log ... log nq k for any α0 ¡ 0 and α1, α2, . . . , αk P R. The sequences used in the proofs of Theorems 1.2.35 and 1.2.33 belong to the class of Toeplitz sequences. A sequence w P XZ is Toeplitz if for every n P Z there exists q P N such that wpn kqq is constant for all k P Z. Note that it follows directly from Proposition 1.2.4 that the closure of the orbit of any Toeplitz sequence is a minimal subshift. It was also shown, see [Fer99, p. 149], that for any 1 α β there exists p pnq p pnq Ñ8 F F a minimal subshift F such that lim infn nα 0 and lim supnÑ8 nβ 8. More on diﬀerent complexity functions of minimal subshifts, see [Fer99] and [CN10].

Proposition 1.2.36. Let F XZ be a subshift. If pF pnq ¤ n for some n, then pF is bounded.

Proof. Suppose that pF pnq ¤ n. By Proposition 1.2.29, the function pF is non-degreasing. We have pF p1q ¡ 1. Consequently, there exists k ¤ n ¡ 1 such that pF pkq pF pk 1q. But this implies, also according to Proposition 1.2.29, that pF is bounded. It follows that the smallest possible unbounded complexity is ppnq n 1. It is realized by the subshifts from the following class. Proposition 1.2.37. Let θ P p0, 1q be an irrational number, and consider the rotation Rθ : x ÞÑ x θ of R{Z. For x P R{Z, denote by Ix P t0, 1uZ the sequence given by the condition " 0 if Rnpxq P r0, 1 ¡ θq; I pnq x 1 if Rnpxq P r1 ¡ θ, 1q, where we identify the circle R{Z with r0, 1q in the natural way. Let Xθ be the closures of the set of sequences Ix for x P r0, 1q. p q Then the complexity of Xθ satisﬁes pXθ n n 1. 36 1. Dynamical systems

The sequences Ix are precisely the sequences considered in 1.1.1, where 0 corresponds to the symbol v, and 1 corresponds to d. Proof. A word v a1a2 . . . an belongs to WFθ if and only if there exists a number x P r0, 1q such that Ixpiq ai for i 1, 2, . . . , n. The set of such points x (for a given v) is an arc of R{Z into which the points 0, 1 ¡ ¡1 ¡2 ¡n θ R p0q,R p0q,...,R p0q subdivide R{Z. The number of such arcs is equal to n 1. In fact, the converse to Proposition 1.2.37 is also true, see Theorem 1.3.38.

1.3. Minimal Cantor systems

We describe in this section theory of minimal Z-actions on the Cantor set via Vershik-Bratteli diagrams. This theory was developed in [HPS92] and found important applications for the theory of orbit equivalence in [GPS95]. For a more detailed exposition, see the book [...]. See also ... Our interest in the subject comes primarily from the fact that minimal Cantor systems provide an interesting class of amenable groups, see... and the literature therein. Moreover, one of the main classes of groups that will be studied in this book can be seen as a direct generalizations of the model of minimal Z-actions via Vershik-Bratteli diagrams, see... A special class of stationary Vershik-Bratteli diagrams provide interesting examples of groups generated by ﬁnite automata and are closely related to hyperbolic dynamical systems.

1.3.1. Examples of minimal homeomorphisms of the Cantor set. 1.3.1.1. Odometers. We have seen (Proposition 1.1.10) that the transformations a ÞÑ a 1 of the ring of dyadic integers Z2 is a minimal homeomorphism. A straightforward generalization of this construction is the transformation x ÞÑ x 1 for an aribtrary proﬁnite completion of Z. More explicitly, consider a sequence of integers greater than one

d1, d2, d3,..., p and the inverse limit Zd1d2... of the cyclic groups

Z{d1Z ÐÝ Z{d1d2Z ÐÝ Z{d1d2d3Z ÐÝ ¤ ¤ ¤ with respect to the natural epimorphisms r d1d2 ¤ ¤ ¤ dnZ ÞÑ r d1d2 ¤ ¤ ¤ dn¡1Z. p The elements of Zd1d2... are uniquely represented as formal expressions

(1.2) r1 r2 ¤ d1 r3 ¤ d1d2 r4 ¤ d1d2d3 ¤ ¤ ¤ , where ri P t0, 1, . . . , di ¡1u. Namely, the formal expression (1.2) corresponds to the element of the inverse limit given by the sequence pr1, r1 r2 ¤ d1, r1 r2 ¤ d1 r3 ¤ d1d2,...q P pZ{d1Z, Z{d1d2Z, Z{d1d2d3Z,...q. 1.3. Minimal Cantor systems 37

The transformation x ÞÑ x 1 can be interpreted as a procedure of adding one to the series (1.2), and then rewriting it in the same form. Namely, if r1 P t0, 1, . . . , d1 ¡2u, then we just replace r1 by r1 1. Otherwise, if r1 d1, we replace r1 by 0, and then change the sequence r2, r3,..., by adding one to the formal series r2 r3 ¤ d2 r4 ¤ d2d3 ¤ ¤ ¤ using the same rule (i.e., replacing r2 by r2 1 if r2 P t0, 1, . . . , d2 ¡2u, etc.). It is easy to see that the same arguments as in the proof of Proposi- ÞÑ p tion 1.1.10 show that the transformation x x 1 of Zd1d2... is a minimal homeomorphism. 1.3.1.2. Irrational rotation. Let us show how one can construct a minimal homeomorphism of a Cantor set from an irrational rotation of the circle. This construction is called sometimes the Denjoy homeomorphism, see...

Let Rθ : x ÞÑ x θ for θ P R r Q be an irrational rotation of R{Z, see 1.1.1. Consider the orbit O tfracpnθq : n P Zu of 0 under Rθ. We represent the points of R{Z by points of r0, 1q (by their fractional parts fracpxq). Let us replace every point α P O r t0u by two copies α 0 and α ¡ 0. We replace 0 P O also by two copies: 0 and 1 (playing the role of 0 0 and 0 ¡ 0, respectively). Consider the obtained set with the natural order: α ¡ 0 α 0; and if α, β P r0, 1s are such that α β, then every copy of α is less than every copy of β. Denote by Xθ the obtained ordered set.

Consider the order topology on Xθ: a basis of topology is the set of all intervals of the form pα, βq, r0, αq, or pα, 1s. We have a natural continuous surjective map Φ : Xθ ÞÑ R{Z mapping each copy of α P r0, 1q to its image in R{Z. The map Φ is at most 2-to-1. One can show that Xθ is homeomorphic to the Cantor set (Exercise 1.19). It is easy to see that R˜θpα 0q Rθpαq 0, R˜θpα ¡ 0q Rθpαq ¡ 0, and R˜θpαq Rθpαq deﬁnes a minimal homeomorphism R˜θ of Xθ. It acts on Xθ as an interval exchange transformation: it moves r0, 1 ¡ θ ¡ 0s to rθ 0, 1s and r1 ¡ θ 0, 1s to r0, θ ¡ 0s by parallel translations. The map Φ is a ˜ semiconjugacy from Rθ y Xθ to Rθ y R{Z.

Proposition 1.3.1. The homeomorphism R˜θ generates an expansive action of Z.

Recall that the homeomorphism Rθ ü R{Z is not expansive, as it is an isometry.

Proof. It is enough to show that for any two points x, y P Xθ such that x y and y ¡ x 1{2 there exists n such that R˜θpxq and R˜θpyq belong to diﬀerent intervals r0, θ ¡ 0s and rθ 0, 1s. But this follows from minimality 38 1. Dynamical systems

of Rθ y R{Z. Namely, if Φpxq Φpyq, then there exists n P Z such that nθ P pΦpxq, Φpyqq, and then θ belongs to the shorter arc of the circle R{Z ¡n 1p p qq ¡n 1p p qq with the endpoints Rθ Φ x and Rθ Φ y , which implies that θ ¡n 1p q ¡n 1p p qq p q p q separates the points Rθ x and Rθ Φ y . If Φ x Φ y , then Φpxq and Φpyq belong to the Rθ-orbit of 0, so there exists n such that ˜ p q ¡ ˜ p q Rθ x θ 0 and Rθ y θ 0.

In fact, it is easy to check that the system R˜θ ü Xθ is topologically conjugate to the subshift Xθ described in Proposition 1.2.37. 1.3.1.3. Minimal subshifts. We have seen many example of minimal Can- tor dynamical systems in Section 1.2, in particular, the ones generated by primitive substitutions. Another important class of minimal subshifts are the Toeplitz subshifts, i.e., subshifts equal to the closure of the set of shifts of a Toeplitz sequence. p q P Z P A sequence xn nPZ X is Toeplitz if for every n Z there exists a positive integer q such that xn qk xn for all k P Z. The closure of the Z-orbit of a Toeplitz sequence under the shift is called a Toeplitz subshift. It easily follows from Proposition 1.2.4 that every Toeplitz subshift is minimal. p q P Z Any Toeplitz xn nPZ X sequence can be constructed in the following way. Let ¦ be a symbol not contained in X, and denote by X¦ X Y t¦u. Suppose that w1, w2 P X¦Z are periodic sequences, and suppose that symbols ¦ p q appear in w1. Denote then by Tw1 w2 the sequence obtained from w1 by replacing consecutive symbols ¦ of w1 by the sequence w2 (so that, for example, the coordinate number 0 of w2 is placed in the ﬁrst non-negative coordinate of w1 equal to ¦). If p1 and p2 are periods of w1 and w2, then p q Tw1 w2 is a periodic sequence of period p1p2. Let w1, w2,... P X¦Z be non-constant periodic sequences with periods p1, p2,... such that ¦ appears in every sequence wi. Suppose that for inﬁn- itely many values of n the symbol ¦ does not appear on the zeroth coordinate p p p p q qqq of wn. Then the sequence Tw1 Tw2 Tw3 ...Twn¡1 wn ... converges to a Toeplitz sequence w P XZ.

Example 1.3.2. Consider the Feigenbaum substitution φ : 0 ÞÑ 11, 1 ÞÑ 10, and let X be the subshift generated by it. Since the words φpxq for x P t0, 1u p q P are of length two and both start with 1, for every sequence xn nPZ X we either have x2n 1 or x2n 1 1 for all n P Z. If we eraze this constant 1 subsequence, the sequence that remains is obtained from a sequence p q P yn nPZ X by changing 0 to 1 and 1 to 0. It follows that every sequence p p p q qq in X is the limit of the sequence Tw1 Tw2 ...Twn¡1 Twn ... , where wn for odd n is one of the two sequences of the form ... ¦ 1 ¦ 1 ¦ 1 ¦ ..., and for even n is one of the two sequences of the form ... ¦ 0 ¦ 0 ¦ 0 ¦ .... 1.3. Minimal Cantor systems 39

The following characterization of Toeplitz shifts shows a relation between Toeplitz subshifts and odometers similar to the relation between Denjoy systems and irrational rotations of the circle.

Theorem 1.3.3 (Downarowicz, Lacroix). A minimal subshift f ü X is Toeplitz if and only if there exists a generalized odometer f 1 ü X 1 and a surjective semiconjugacy π : X ÝÑ X 1 such that |π¡1pxq| 1 for some x P X 1.

1.3.2. Rokhlin-Kakutani towers.

Lemma 1.3.4. Let f ü X be a minimal homeomorphism of a compact topological space. For every open set U X and every x P X there exist n n¡ integers n ¡ 0 and n¡ ¤ 0 such that tf pxq, f pxqu U.

n Proof. By minimality, the sets f pUq, for n P Z, cover X . By compactness, n there exists a ﬁnite set n1, n2, . . . , nk P Z such that the sets f i pUq cover X . ¡ max n m Applying f i to this cover, we get a cover tf i pUqui1,...,k where all ¡ min n 1 the numbers mi are non-positive. Similarly, applying f i , we get a t mi p qu cover f U i1,...,k where all the numbers mi are positive.

Let τ ü X be a minimal homeomorphism of a compact totally disconnected metrizable space. A Rokhlin-Kakutani partition is a collection of ﬁnite sequences

k1¡1 pC1, τpC1q, τpC1q, . . . , τ pC1qq,

k2¡1 pC2, τpC2q, τpC2q, . . . , τ pC2qq, . .

km¡1 pCm, τpCmq, τpCmq, . . . , τ pCmqq t ip q ¤ u of clopen subsets of X such that τ Cj : 0 i ki is a parition of X , ki p q m and τ Ci j1 Cj. Each sequence

2 ki¡1 pCi, τpCiq, τ pCiq, . . . , τ pCiqq m is called a tower of the partition. The set i1 Ci is called the base of the partition, and the set Ci is called the base of the corresponding tower. See Figure 1.14 where the action of τ on a Rokhlin-Kakutani partition is shown schematically.

Proposition 1.3.5. For every ﬁnite clopen partition P of X and every clopen subset C X there exists a Rokhlin-Kakutani parition subordinate to P with the base equal to C. 40 1. Dynamical systems

Figure 1.14. A Rokhlin-Kakutani partition

Proof. By Lemma 1.3.4, for every x P X there exists a positive integer n n n¡ and a non-positive integer n¡ such that τ pxq, τ P C. Let n pxq and n¡pxq be the smallest and the largest of such numbers, respectively. They are the ﬁrst return times to C of the dynamical system. Note that for every k P N the set of points tx P X : n pxq ku is equal to ¤ ¡k ¡i τ pCq r τ pCq. 1¤i k Similarly, the set tx P X : n¡pxq ¡ku is equal to ¤ k i τ pCq r τ pCq. 0¤i k Note that since C is clopen and τ is a homeomorphism, these sets are clopen. It follows that the functions n pxq and n¡pxq are locally constant, i.e., continuous. Consequently, the map x ÞÑ pn¡pxq, n pxqq is locally constant, hence its set of values is ﬁnite (as X is compact). Denote by

Ci,j tx P X : n¡pxq ¡i, n pxq ju its level sets. They form a ﬁnite clopen partition of X . Note that C is equal to the set of points x P X such that n¡pxq 0. It follows that C is partitioned into a ﬁnite collection of disjoint sets

C0,k1 ,C0,k2 ,,...,C0,km for some positive integers ki. 1.3. Minimal Cantor systems 41

Figure 1.15. A Bratteli diagram

For every ki and 1 ¤ l ki we obviously have lp q τ C0,ki Cl,ki¡l, lp q since the first moment in the future when points of τ C0,ki come back to C, by definition, is ki ¡ l (as the first positive return time to C of points of lp q C0,ki is ki); and the first time in the past when points of τ C0,ki are in C is ¡l. It follows that the sets Ci,j form a Rokhlin-Kakutani partition with base C and towers

C0,ki ,C1,ki¡1,C2,ki¡2,,...,Cki¡1,1. P p q For a point x C0,ki consider the itinerary P0,P1,...,Pki¡1 of x with l respect to P, i.e., a sequence of elements of P such that τ pxq P Pl for 0 ¤ l ¤ ¡ p q ki 1. Since P is finite, the set of all itineraries P0,P1,...,Pki¡1 is finite, hence C0,ki is partitioned into a finite set of clopen subsets A1,A2,...,AMi such that all points one set Aj have the same itineraries. Let us split the tower into Mi towers

2 ki¡1 Aj, τpAjq, τ pAjq, . . . , τ pAjq. Then each element of the tower is a subset of an element of P. The union of all such towers (for all towers of the partition tCi,ju) will be a Rokhlin- Kakutani partition subordinate to P and with the base equal to C. 1.3.3. Bratteli diagrams. A Bratteli diagram B consists of sequences p q p q ÝÑ V1,V2,... and E1,E2,... of ﬁnite sets and sequences of maps sn : En ÝÑ Vn, rn : En Vn 1. The sets V n¥1 Vn and E n¥1 En are the sets of vertices and edges of the diagram, respectively. An edge e P En connects the vertices snpeq P Vn and rnpeq P Vn 1. See Figure 1.15 where a beginning of a Bratteli diagram is shown.

We assume that the maps sn and rn are surjective (though more general Bratteli diagrams are also sometimes considered in the literature). We will sometimes denote s sn and r rn when the domains of the corresponding maps are clear from the context. 42 1. Dynamical systems

Let us enumerate each set Vn, i.e., identify it with t1, 2,..., |Vn|u. The the Bratteli diagram is determined then by the sequence of matrices An p q P mij 1¤i¤|Vn 1|,1¤j¤|Vn|, where mij is the number of edges connecting i Vn 1 to j P Vn. For example, if we enumerate the vertices of the levels of the diagram shown on Figure 1.15 from left to right, then we have ¤ ¢ 1 0 ¢ 2 0 1 1 3 0 A ,A ¥ 1 1 ,A . 1 0 1 0 2 3 0 0 1 0 1 A (finite or infinite) path in the diagram B is a (finite or infinite) sequence pe1, e2,...q, where en P En and rnpenq sn 1pen 1q for all n. If pe1, e2, . . . , enq is a finite path, then its length is n. Additionally, a path of length 0 is a vertex of V1. We will also sometimes consider paths from Vi to Vj for arbitrary i j, but then we explicitly specify that the path begins in a vertex of Vi.

It is easy to see that the number of paths from i P V1 to j P Vn is equal to the entry in the ith column and jth row of the matrix A1A2 ¤ ¤ ¤ An¡1.

We denote by PnpBq the set of paths of length n (from V1 to Vn). The set of all inﬁnite paths PpBq is a closed subset of the direct product E1¢E2¢¤ ¤ ¤ , and we consider it as a topological space with the induced topology. The space PpBq is compact, totally disconnected, and metrizable.

We say that a path pe1, e2, . . . , enq P PnpBq starts in s1pe1q and ends in rnpenq. The multiplicity of a vertex v P Vn is the number of paths γ P PnpBq ending in v. We denote it mpvq. pp q8 p q8 q 9 If B Vn n1, En n1, s, r is a Bratteli diagram, then we denote by B pp q8 p q8 q the diagram Vn n0, En n0, s, r obtained by adding one level of vertices V0 consisting of one vertex and one level of edges E0 such that r : E0 ÝÑ V1 is a bijection. pp q8 p q8 q Deﬁnition 1.3.6. Let B Vn n0, En n0, s, r be a Bratteli diagram. Let k0 0 k1 k2 ... be an increasing sequence of integers. The tele- pp 1q8 p 1 q8 q scoping of B deﬁned by the sequence is the diagram Vn n0, En n0, s, r , 1 1 where Vn Vkn , En is the set of paths in B from Vkn to Vkn 1 , and s, r are the beginning and the end of the paths, respectively.

We write B1 B2 (and call the diagrams B1 and B2 equivalent) if one 9 can transform B1 by a sequence of telescopings and operations inverse to 9 telescoping to a diagram isomorphic to B2. We will see later that B1 B2 if and only if there exists a diagram B such that a telescoping of B9 is a 9 9 9 telescoping of B1 and another telescoping of B is a telescoping of B2.

Example 1.3.7. Let B1 be the diagram with one vertex and two edges on every level. Let B2 be the diagram with two vertices on each level and 1.3. Minimal Cantor systems 43

Figure 1.16. Equivalence of diagrams complete bipartite graphs of edges on each level. Then they are equivalent, see Figure 1.16, where telescopings of the diagram B shown in the middle 9 9 are isomorphic to the diagram B1 shown on the left and to the diagram B2 shown on the right.

Bratteli diagrams were introduced by O. Bratteli in [Bra72] to describe approximately ﬁnite C¦-algebras (i.e., direct limits of ﬁnitely dimensional C¦-algebras). They can be used to describe inductive limits of direct products of various algebraic structures with respect to block-diagonal embeddings.

ExampleÀ 1.3.8. Consider for every level n of the diagram B the direct sum A M p q¢ p qp q of the algebras of mpvq ¢ mpvq-matrices over a n vPVn m v m v k ﬁeld k. For every vertex v P Vn consider the set of edges e P En¡1 ending in p q v. Then the algebra MmÀpvq¢mpvq k contains a sub-algebra of block-diagonal p q matrices isomorphic to ePr¡1pvq Mmpspeqq¢mpspeqq k . We get hence a block- diagonal embeddings An¡1 ãÑ An, and the corresponding inductive limit MBpkq, deﬁned by the diagram. We can also consider the inductive limit ¦ in the category of C -algebras, in the case k C, which was the original motivation of [Bra72]. 44 1. Dynamical systems

¥ Example± 1.3.9. Let G be a group. For every n 1, let Gn be the direct mpvq mpvq product G . For every v P Vn the group G is isomorphic to ± vPVn mpspeqq ePr¡1pvq G , where the direct factors are labeled by paths of length n. We have natural block-diagonal embeddings Gn¡1 ãÑ Gn mapping the factor corresponding to a path γ diagonally to the factors corresponding to its continuations. The inductive limit of these maps is naturally isomorphic to the group of all continuous (i.e., locally constant) maps PpBq ÝÑ G with pointwise multiplication, where G has discrete topology.

Example 1.3.10. Consider the direct sums Gn of the symmetric groups Smpvq for v P Vn. Each group Smpvq acts by permutations on the set of all paths γ P PnpBq ending in v. Then the permutation group Smpvq naturally contains an isomorphic copy of the direct sum of the permutation ¡1 groups Smpspeqq for all e P r pvq. We get hence block-diagonal embeddings Gn¡1 ãÑ Gn and the direct limit of finite groups, defined by the diagram. Similarly, one can take direct sums of the alternating groups Ampvq and the corresponding embeddings and direct limit. See 5.2.3 for more about these constructions. Example 1.3.11. Let B be a Bratteli diagram defined by the sequence of matrices A1,A2,.... Consider the sequence of abelian groups

|V | A1 |V | A2 |V | A3 Z 1 ÝÑ Z 2 ÝÑ Z 3 ÝÑ ¤ ¤ ¤ . Its direct limit is called the dimension group of the diagram. Its positive |Vn| |V | cone is the union of the images of the subsemigroups Z Z n in the direct limit (where Z is the semigroup of non-negative integers).

Note that in all the above examples B1 B2 implies for each of Exam- ples 1.3.8–1.3.11 that the direct limits deﬁned by the diagrams B1 and B2 are isomorphic.

1.3.4. Vershik-Bratteli diagrams. Deﬁnition 1.3.12. A Vershik-Bratteli diagram (or an ordered diagram) is a Bratteli diagram together with a linear order on each set r¡1pvq. An edge e is called minimal (resp. maximal) if it is minimal (resp. maximal) in the set r¡1prpeqq. A path is said to be minimal (resp. maximal) if it consists of minimal (resp. maximal) edges only.

An ordering of the edges of a Vershik-Bratteli diagram B ppViq, pEiq, psiq, priqq deﬁnes a natural lexicographic order on the sets of path between Vi and Vj for every pair i j and on the set PpBq of inﬁnite paths in B.

Namely, two ﬁnite paths pe1, e2, . . . , enq, pf1, f2, . . . , fnq are comparable if and only if rpenq rpfnq. Then pe1, e2, . . . , enq pf1, f2, . . . , fnq if ek fk 1.3. Minimal Cantor systems 45

for the largest index k such that ek fk. Note that then rpekq rpfkq, so ek and fk are comparable. In particular, a telescoping of a Vershik-Bratteli diagram is again a Vershik-Bratteli diagram with respect to the lexicographic order on the contracted paths (which become edges of the telescoping, see Deﬁnition 1.3.6).

Similarly, two infinite paths pe1, e2,...q are comparable if and only if they are co-final, i.e., if en fn for all n big enough. Then we have pe1, e2,...q pf1, f2,...q if ek fk for the largest index k such that ek fk. The adic transformation (or the Vershik map) is defined on the set of non-maximal paths in PpBq and maps a path γ pe1, e2,...q to the next path in the lexicographic order (i.e., to the smallest path among the paths bigger that γ). It is computed in the following way. Find the first in- 1 dex n such that the edge en is non-maximal. Let en be the next edge ¡1p p qq p 1 1 1 q in rn rn en , and let e1, e2, . . . , en¡1 be the minimal path such that p 1 q p 1 q rn¡1 en¡1 sn en (it exists and is unique, since for every vertex v there exists a unique minimal edge e P r¡1pvq). It is easy to see that the path p 1 1 1 1 q e1, e2, . . . , en¡1, en, en 1, en 2,... is the minimal path among the paths bigger than γ in the lexicographic order. See Figure 1.17, where the arrows show the ordering of the edges, and point from a smaller edge to the bigger one. The map τ changes the highlighted red beginning of a path to the black path. (check the colors...) The adic transformation is defined on the set of non-maximal paths, and its set of values is the set of non-minimal paths. Note that the inverse of the adic transformation is the adic transformation of the Vershik-Bratteli diagram obtained from B by reverting the ordering of the edges. The adic transformation is continuous on the set of non-maximal paths, since it changes in every non-maximal path pe1, e2,...q only the finite beginning pe1, e2, . . . , enq, where en is the first non-maximal edge in the path. However, it may not have a continuous extension to the whole space PpBq. Consider, for example, the Vershik-Bratteli diagram shown on the right- hand side of Figure 1.25. Let us label the vertices of each level by 0 and 1 so that 0 is on the left-hand side, and denote paths in the diagram by the sequence of vertices through which it passes (in this case this notation is non-ambiguous). It has two minimal paths 010101 ... and 101010 ..., and one maximal path 1111 .... The adic transformation can not be extended continuously to the whole space of paths. Namely, it maps a sequence w 111 ... 101v either to 0101 ... 011v or to 1010 ... 011v depending on the parity of the number of the leading ones in w, so it can not be continuously defined at 111 .... Another example is given by the diagram shown on the right-hands side half of Figure 1.19. In this case there are two maximal and two minimal 46 1. Dynamical systems

Figure 1.17. Adic transformation inﬁnite paths, but there is no continuous extension of the adic transformation to the whole space of paths. Deﬁnition 1.3.13. We say that a Vershik-Bratteli diagram B is properly ordered if it has a unique maximal and a unique minimal paths in PpBq. Proposition 1.3.14. Suppose that a Vershik-Bratteli diagram B is properly ordered. Let γmax and γmin be the maximal and the minimal paths, respectively. Then the extension of the adic transformation τ given by τpγmaxq γmin is a homeomorphism. We will also call the extension of the adic transformation τ given in Proposition 1.3.14 the adic homeomorphism.

Proof. Since changing the ordering to the opposite one changes the adic transformation to the inverse, it is enough to prove that the deﬁned extension of the adic transformation is continuous. The transformation is continuous at non-maximal paths, hence it is enough to prove that τ is continuous at γmax.

Let γi be a sequence of non-maximal inﬁnite paths converging to γmax. It is enough to prove that we always have limiÑ8 τpγiq γmin. Let αi be 1.3. Minimal Cantor systems 47

the longest common prefix of γi and γmax. Let ni be the length of αi. We have ni Ñ 8 as i Ñ 8. By the definition of τ, the beginning of τpγiq of length ni is a finite minimal path (i.e., a finite path consisting of minimal edges only). It follows that the limit limiÑ8 γi is a minimal infinite path, hence it is equal to γmin. Definition 1.3.15. We say that a Bratteli diagram B is simple if for every level Vn there exists m ¡ n such that for every pair v P Vn and u P Vm there exists a path pen, en 1, . . . , em¡1q in B starting in v and ending in u. Proposition 1.3.16. Let B be a properly ordered Vershik-Bratteli diagram, and let τ ü PpBq be the corresponding adic homeomorphism. There exists a unique closed non-empty subset Y PpBq such that τ ü Y is a minimal homeomorphism. If B is simple, then τ ü PpBq is minimal, i.e., Y PpBq.

Proof. Let γ P PpBq be an arbitrary path, and consider a finite beginning α of γ. By the definition of the adic transformation, there exists k ¥ 0 such that τ kpγq begins by the maximal path in the lexicographic order among the paths ending in the same vertex as α. It follows that the forward orbit τ npγq for n ¥ 0 contains arbitrarily long prefixes that are maximal paths (it is possible that one of such prefixes is the whole path γmax). It follows from the uniqueness of the maximal path that γmax belongs in the closure of the τ-orbit of γ. Consequently, closure of the τ-orbit of γmax is the unique closed non-empty subset Y PpBq such that τ ü Y is minimal. If B is simple, then for any infinite path γ and for any finite path α there exists a finite path β such that αβ ends in a vertex of γ. It follows that there exists a path γ1 P PpBq starting with α that has a common infinite suffix with γ. Then γ1 and γ belong to the same τ-orbit. This shows that the τ-orbit of γ intersects every open subset of PpBq, i.e., that every τ-orbit is dense.

Example 1.3.17. Let d1, d2,... be a sequence of integers greater than 1. Consider the associated odometer x ÞÑ x 1 acting on the inverse limit of the cyclic groups Z{d1d2 ¤ ¤ ¤ dnZ, see 1.3.1.1. It follows from the description of the action of the odometer on the set of inﬁnite formal series (1.2) that it is topologically conjugate to the adic homeomorphism deﬁned by the Vershik- Bratteli diagram B such that |Vn| 1 and |En| dn. A particular case, for 2 d1 d2 ..., was considered in 1.1.4.

1.3.5. Diagrams associated with sequences of Rokhlin-Kakutani partitions. Let τ ü X be a minimal homeomorphism of a Cantor set. If R1 and R2 are Rokhlin-Kakutani partitions, then we write R1 R2 if the base of R1 is contained in the base of R2, and R2 is a reﬁnement of R1 (i.e., every element of R2 is a subset of an element of R1). 48 1. Dynamical systems

Consider a sequence of Rokhlin-Kakutani paritions R1, R2,... with bases B1,B2,... such that R1 R2 .... We assume that the base B1 is equal to X , i.e., that all towers of R1 are of length 1. We are going to associate a Vershik-Bratteli diagram B with the sequence Rn. Take the nth level set of vertices Vn equal to the set of towers of Rn. (In particular, V1 is the set of elements of R1.) k¡1 Let pC, τpCq, . . . , τ pCqq be a tower of Rn 1. Then C Bn 1 is a 1 1 subset of an element C of Rn, and C Bn. Each of the elements of the tower is also contained in an element of Rn. It follows that the tower is naturally split into segments

l ¡1 pC0 CC, τpCq, . . . , τ 1 pCqq, l l 1 l l ¡1 pC1 τ 1 pCq, τ 1 pCq, . . . , τ 1 2 pCqq, l l l l 1 l l l ¡1 pC2 τ 1 2 pCq, τ 1 2 pCq, . . . , τ 1 2 3 pCqq, . . l l ¤¤¤ l ¡ l l ¤¤¤ l ¡ 1q l l ¤¤¤ls¡1 pCs¡1 τ 1 2 s 1 pCq, τ 1 2 s 1 pCq, . . . , τ 1 2 pCqq, such that for each segment

li 1¡1 pCi, τpCiq, . . . τ pCiqq p p q p qq there exists a (necessarily unique) tower vi Ci, τ Ci , . . . , τli¡1 Ci of Rn such that l1 l2 ¤¤¤ li¡1 j j τ pCq τ pCiq.

In other words, the towers of Rn 1 are split into disjoint unions of restric- tions of towers of Rn. Let us connect the vertex of Vn 1 corresponding to k¡1 the tower pC, τpCq, . . . , τ pCqq of Rn 1 to the vertices corresponding to the towers vi in the natural order of their appearance in the above list of segments. Note that vi are not necessarily pairwise diﬀerent, so we may get multiple edges. Then each edge connecting a tower pA, τpAq, . . . , τ kpAqq l of Rn 1 to a tower pB, τpBq, . . . , τ pBqq of Rn is in a bijective correspondence with a segment pτ ipAq, τ i 1pAq, . . . , τ i lpAqq of the ﬁrst tower such that τ ipAq B. We say that this segment corresponds to the edge. We call the constructed Vershik-Bratteli diagram B the diagram associated with the sequence pRnqn1,2,....

See Figure 1.18, where an example of a pair Rn Rn 1 of Rokhlin- Kakutani partitions and the corresponding level of the Vershik-Bratteli diagram are shown. Bigger boxes represent elements of Rn. Black and gray rectangles depict elements of the partitions Rn 1 and Rn, respectively, belonging to their bases. The towers and the corresponding vertices of the Vershik-Bratteli diagram are labeled by letters vi and ui are placed near the elements belonging to the bases. 1.3. Minimal Cantor systems 49

Figure 1.18. Reﬁnement of a Rokhlin-Kakutani partition

Proposition 1.3.18. Let B be the Vershik-Bratteli diagram associated with a sequence R1 R2 .... Then there is a unique sequence of bijections φn : PnpBq ÝÑ Rn such that φnpe1, e2, . . . , enq P Rn is a subset of φn¡1pe1, e2, . . . , en¡1q (of φ0pspe1qq for n 1), and an element of the segment en of the tower rpenq.

Proof. Let e P En be an edge of B. Then speq is a tower of Rn, and rpeq is a tower of Rn 1. The tower rpeq contains a segment (determined by e) A, τpAq, . . . , τ l¡1pAq such that the tower speq is of the form A1, τpA1q, . . . , τ ¡1pA1q 1 1 1 for some A P Rn such that A A . We see that for every element C of the tower speq there is a unique element C of the tower rpeq such that C C1 and C belongs to the segment fo rpeq corresponding to the edge e. The proof of the proposition now follows by induction. The following proposition is also proved by induction on n. We leave it as an exercise.

Proposition 1.3.19. Two paths γ1, γ2 P PnpBq are comparable if and only if φnpγ1q and φnpγ2q belong to the same tower of Rn. If γ2 is the smallest path bigger than γ1, then τpφnpγ1qq φnpγ2q.

In particular, a path γ P PnpBqq is minimal (resp. maximal) if and only if φnpγq (resp. τpφnpγqq) is contained in the base of Rn. The following theorem is a result of [HPS92]... Theorem 1.3.20 (R.H. Herman, I.F. Putnam, C.F. Skau). Let τ ü X be a minimal homeomorphism of a Cantor set. Then it is topologically conjugate to the adic homeomorphism of a simple properly ordered Vershik-Bratteli diagram.

Proof. Let f ü X be a minimal homeomorphism of a Cantor set. Choose a metric d on X compatible with the topology. Take an arbitrary point x0 P X . 50 1. Dynamical systems

Let R0 be an arbitrary finite partition of X into non-empty clopen subsets (e.g., tX u), seen as a Rokhlin-Kakutani partition with the base X . Let Pn be a sequence of finite clopen partitions such that maximum Dn of diameters of elements of Pn converges to zero. Let Cn P Pn be the element containing x0. Construct recursively Rn as a Rokhlin-Kakutani partition subordinate to Rn¡1 and Pn with the base a subset of Cn, see Proposition 1.3.5. We get a sequence R0 R1 .... Let B be the associated Vershik-Bratteli diagram. Let φn : PnpBq ÝÑ Rn be the bijections described in Proposition 1.3.18. Then for every infinite path γ pe1, e2,...q P PpBq we get a decreasing sequence of sets φnpe1, e2, . . . , enq P Rn. Their intersection is non-empty by compactness, and of diameter zero, since the diameter of every element of Rn is less than Dn. It follows that they intersect in one point, which we will denote φpγq. If γ1 and γ2 agree on a beginning of length n, then dpφpγ1q, φpγ2qq ¤ Dn, which implies that φ : PpBq ÝÑ X is continuous. For every x P X and n the point x belongs to a single element of Rn, and, by Proposition 1.3.18, this element is equal to φnpe1, e2, . . . , enq for some γ pe1, e2,...q P PpBq not depending on n. It follows that φ is onto. It is also easy to see that it is one-to-one, hence a homeomorphism.

The intersection of the bases of Rn is non-empty (since the base of Rn 1 is non-empty and contained in the base of Rn) and contained in Cn. It follows that the intersection of the bases is x0. It follows that the minimal path in B is unique and its image under φ is x0 (see the characterization of the minimal path in Proposition 1.3.18). Similarly, the intersection of ¡1 ¡1 the images of the bases under f is equal to tf px0qu. In particular, the minimal and the maximal paths in B are unique. Proposition 1.3.18 shows now that φ conjugates the adic homeomorphism with f.

In fact, the proof of Theorem 1.3.20 shows if τ ü X is a minimal system, and x P X , then there exists a Vershik-Bratteli diagram B and a homeomorphism φ : PpBq ÝÑ X satisfying the conditions of the theorem and such that φ maps the minimal path to x. In fact, we have the following characterization of the Vershik-Bratteli models of minimal Cantor systems.

Theorem 1.3.21. Let B1 and B2 be properly ordered Vershik-Bratteli diagrams, and let τi be the adic homeomorphism of PpBiq. Then the Vershik- Bratteli diagrams Bi are equivalent if and only if there exists a homeomorphism φ : PpB1q ÝÑ PpB2q mapping the minimal, resp. maximal, path of B1 to the minimal, resp. maximal, path of B2 and conjugating the corresponding adic transformations.

reference... 1.3. Minimal Cantor systems 51

Proof. (A sketch.) Telescoping a Vershik-Bratteli diagram does not change the space of its paths and the lexicographic order on them, therefore it does not change the corresponding adic transformation. It follows that equivalent Vershik-Bratteli diagrams have topologically conjugate adic transformations.

In the other direction, it is easy to see that if R0 R1 R2 ... is an increasing sequence of Rokhlin-Kakutani partitions with the associated Vershik-Bratteli diagram B, then the diagram associated with a subsequence

R0 Rn1 Rn2 ... is a telescoping of the diagram B. Suppose that 1 1 R0 R1 R2 ... and R0 R1 R2 are two sequences of Rokhlin- Kakutani partitions for a given minimal system f ü X such that the maximal diameters of the elements of the partitions go to zero in both sequences, P 1 and x0 X is contained in the bases of all partitions Rn and Rn. Then it is easy to prove (using Lebesgue lemma) that for every n there 1 1 exists m such that Rn Rm and Rn Rm. It follows that there exists a 1 1 sequence n1 n2 n3 ... such that R0 Rn1 Rn2 Rn3 .... This shows that the associated Vershik-Bratteli diagrams of the sequences pRnq p q1 and Rn are equivalent.

1.3.6. Kakutani equivalence. Let f ü X be a minimal homeomorphism of a Cantor set, and let Y X be a non-empty clopen subset. Then for every x P Y there exists n ¡ 0 such that f npxq P Y, see Lemma 1.3.4. nx Let nx be the smallest such number. Then fY : x ÞÑ f pxq is called the first return map to Y. Since x ÞÑ nx is locally constant, it is continuous. mx The inverse map is the map x ÞÑ f pxq, where mx is the smallest positive ¡ integer such that f mx pxq P Y. Since the orbits of the first return map are intersections of the f-orbits with Y, the first return map fY ü Y is a minimal homeomorphism.

Deﬁnition 1.3.22. Two minimal Cantor dynamical systems f1 ü X1 and f2 ü X2 are said to be Kakutani equivalent if there exist non-empty clopen p q ü p q ü subsets Yi Xi such that ﬁrst return maps to f1 Y1 Y1 and f2 Y2 Y2 are topologically conjugate.

Proposition 1.3.23. The Kakutani equivalence is an equivalence relation.

Proof. The Kakutani equivalence is obviously reﬂexive and symmetric. Let us show that it is transitive. Suppose that f1 ü X1 is Kakutani equivalent to f2 ü X2, and f2 ü X2 is equivalent to f3 ü X3. Then there exist non-empty clopen subsets Y1 X1, Y2 X2, Z2 X2, Z3 X3 such that p q ü p q ü p q ü f1 Y1 Y1 is topologically conjugate to f2 Y2 Y2 and f2 Z2 Z2 is p q ü topologically conjugate to f3 Z3 Z3. 52 1. Dynamical systems

m There exists m such that U f pZ2q X Y2 is non-empty. Then the ÝÑ p q ü p q conjugacy φ1 : Y1 Y2 between f1 Y1 f2 Y2 restricts to a topological ¡1 ¡m conjugacy of pf1qφ¡1pUq ü φ pUq with pf2qU ü U. The map f : X2 ÝÑ X2 is a conjugacy of f2 ü X2 with itself, which induces a conjugacy of ¡m ¡m pf2qU ü U with pf2qf ¡mpUq ü f pUq. Note that f pUq Z2, and then ÞÑ p q ü p q ü the conjugacy φ2 : Z2 Z3 of f2 Z2 Z2 with f3 Z3 Z3 will induce a ¡m ¡m p q ¡m ü p q p q ¡m ü p p qq conjugacy of f2 f pUq f U with f3 φ2pf pUqq φ2 f U . It ¡m p q ü p q ¡m ü p p qq follows that f1 U U and f3 φ2pf pUqq φ2 f U are topologically conjugate, hence f1 and f3 are Kakutani equivalent.

Example 1.3.24. Consider the Denjoy homeomorphism R˜θ ü Xθ described in 1.3.1.2. Let a, b P r0, 1s be elements of the orbit of 0 under the rotation Rθ. Suppose that a b, and let ra 0, b ¡ 0s be the corresponding clopen subset of Xtheta. We will sometimes denote such subsets just ra, bs, when it does not lead to confusion. Let n be the smallest positive integer such ˜ that c fracpb nθq P pa, bq. Then the ﬁrst return map pRθqra,bs maps the interval rb¡c a, bs to ra, cs and the interval ra, b¡c as to rc, bs by parallel translations. If we identify ra, bs with r0, 1s by the aﬃne transformation ÞÑ x¡a b¡c b¡c x b¡a , then we get the transformation swapping b¡a , 1 with 0, b¡a , b¡c b¡c kθ l i.e., the rotation by the angle b¡a . Note that b¡a is of the form mθ n for some k, l, m, n P Z, since a, b, c belong to the orbit of 0 under the rotation by θ. We will give a complete description of the Kakutani equivalence classes of the homeomorphisms R˜θ later.

Proposition 1.3.25. Let B1 and B2 are properly ordered Vershik-Bratteli diagrams. The associated adic transformations are Kakutani equivalent if and only if the diagrams B1 and B2 are equivalent to Vershik-Bratteli dia- 1 1 grams B1 and B2 that diﬀer from each other on a ﬁnite number of levels.

Proof. Let R1 R2 ... be a sequence of Rokhlin-Kakutani partitions, 1 and let B be the associated Vershik-Bratteli diagram, as in in 1.3.2. Let Rn for n ¥ 2 be the partition of the base of R2 consisting of the elements of 1 1 Rn contain in it. Then R2 R3 ... is a sequence of Rokhlin-Kakutani partitions of the first return map to the base of R2. It is also easy to check p 1 q that the Vershik-Bratteli diagram associated with the sequence Rn n¥2 is obtained by deleting the first level of the diagram B. It follows that remov- ing a finite number of initial levels of a Vershik-Bratteli diagram does not change the Kakutani class of the associated adic homeomorphism. Conse- quenlty, any finite change in the Vershik-Bratteli diagram does not change the Kakutani class of the adic transformation, since any such a change can be erased by deleting a finite number of levels. This proves the “if” direction of the proposition. 1.3. Minimal Cantor systems 53

Let f ü X be a minimal homeomorphism, let Y X be a non-empty clopen subset, and let R0 R1 R2 ... be a sequence of Rokhlin- Kakutani partitions of f ü X separating the points of X , so that the adic transformation on the associated Vershik-Bratteli diagram is naturally conjugate to f ü X . We assume that R0 tX u. We may also assume that the base of R1 is contained in Y and every element of R1 is either contained in Y or disjoint with it. Then the same conditions will be satisfied for all Rn, n ¥ 1 (by the definition of the relation “ ” on Rokhlin-Kakutani par- 1 ¥ titions). Denote by Rn, for n 1, the partition of Y equal to set of the 1 elements of Rn that are contained in Y. It is checked directly that Rn is a Rokhlin-Kakutani partition of the first return map fY ü Y, so that we get sequence of Rokhlin-Kakutani partitions

tX u R1 R2 R3 ... and t u 1 1 1 Y R1 R2 R3 ... of the sysetms f ü X and fY ü Y, respectively. Both sequences separates the points of the corresponding spaces, so the associated Vershik-Bratteli diagrams model the corresponding dynamical systems. These Vershik-Bratteli diagrams diﬀer only on the ﬁrst level. This proves the “only if” direction of the proposition.

1.3.7. Vershik-Bratteli diagrams as sequences of substitutions. Ev- ery level pVn,En,Vn 1q of a Vershik-Bratteli diagram naturally deﬁnes a ho- ¦ ÝÑ ¦ P momorphism of monoids φn : Vn 1 Vn . Namely, if v Vn 1, and if ¡1 pe1, e2, . . . , emq is the set r pvq listed according to the ordering of En, then we set φnpvq spe1qspe2q ... spemq. Conversely, every sequence of monoid homomorphisms

¦ ÐÝφ1 ¦ ÐÝφ2 ¦ ÐÝφ3 ¤ ¤ ¤ (1.3) V1 V2 V3 is naturally encoded by the Vershik-Bratteli diagram with the sets of ver- ¡1 tices V1,V2,..., where for every vertex v P Vn 1 the set r pvq of edges ending in v has |φpvq| elements e1 e2 ... e|φpvq|, and φpvq spe1qspe2q ... spe|φpvq|q. Telescoping of a Vershik-Bratteli diagram corresponds in this interpretation to compositions of the homomorphisms, i.e., replacing ¦ the sequence (1.3) of monoids by a subsequence (containing V1 ) connected by the corresponding compositions of morphisms.

Deﬁnition 1.3.26. Consider a sequence

¦ ÐÝφ1 ¦ ÐÝφ2 ¦ ÐÝφ3 ¤ ¤ ¤ X1 X2 X3 54 1. Dynamical systems

Z of substitutions. The subshift generated by it is the subshift F X1 of p q all sequences xn nPZ such that for every ﬁnite subword v xnxn 1 . . . xm there exists k and x P Xk such that v is a subword of φ1 ¥ φ2 ¥ ¤ ¤ ¤ ¥ φkpxq.

If the diagram is stationary, i.e., if all sequences Vn,En, sn, rn and the ordering are constant, then the sequence φn is also constant. Conversely, the Vershik-Bratteli diagram associated with a constant sequence pφ, φ, . . .q of endomorphisms φ : X¦ ÝÑ X¦ is stationary. For example, the stationary Vershik-Bratteli diagram on the left-hand side part of Figure 1.25 is associated with the substitution σp0q 01, σp1q 011, if we label the vertices of a level of the diagram by 0 and 1 from left to right. Let B be a Vershik-Bratteli diagram, and consider the corresponding ¦ ÝÑ ¦ P sequence φn : Vn 1 Vn of monoid homomorphisms. For every v Vn the set of paths ending in v is in a natural bijective order-preserving correspondence with the letters of the word φ1 ¥ φ2 ¥ ¤ ¤ ¤ ¥ φn¡1pvq. Let p q p q γ e1, e2,... be an inﬁnite path in B. Denote wn γ xk1 xk1 1 . . . xk2 φ1 ¥ φ2 ¥ ¤ ¤ ¤ ¥ φnprpenqq, where the numbering of the letters is by consecutive integers such that x0 is the letter corresponding to the path pe1, e2, . . . , enq. Then wn 1pγq is obtained from wnpγq by appending letters to the beginning and/or to the end of wnpγq. It follows that in the limit we get a word w8pγq associated with the path γ. The adic transformation acts as the shift on the associated words. Note that if γ is a minimal path, then w8pγq is of the form x0x1 .... If γ is maximal, then w8pγq is of the form . . . x¡1x0. In general, the word w8pγq may be ﬁnite.

Different paths γ may produce the same words. For example, if Vn txu and |En| 2 for every n, then infinitely many paths in the diagram will be associated with the word . . . xxxx . . .. The remaining paths will be associated with one-sided sequences xxxx . . . and . . . xxxx. In order to recover the path γ from the word w8pγq, we have to remember the “production process” of the words wn and position of the zeroth coordinate. For example, in the last example, we can put brackets around subwords equal to the images of the single letters x P Vn under compositions of the homomorphisms. Such a bracketing may be still not enough for some Vershik-Bratteli diagrams, for example, if images of the different letters are equal. (Take, for example, the Vershik-Bratteli diagram of the constant sequence of substitutions σpaq ab, σpbq ab.) The problem of uniqueness of the path γ for a word w8pγq is called recognizability, see [Ku03, Section 4.3]. Example 1.3.27. Let σ : 0 ÞÑ 01, σ : 1 ÞÑ 10 be the Thue-Morse substi- p q tution. Note that if w xn nPZ is an element of the subshift generated 1.3. Minimal Cantor systems 55

by it, and xnxn 1 00 or xnxn 1 11, then we know that xn¡1xn and xn 1xn 2 are images of letters under σ. After that the preimage of w is uniquely determined. Note that we always can ﬁnd such an index n, since otherwise the word w is of the form . . . abababab . . ., which does not belong to the Thue-Morse subshift. Consequently, the word w uniquely determines the corresponding path in the Vershik-Bratteli diagram of the substitution. Example 1.3.28. Similarly, consider the Fibonacci substitution σp0q 01, σp1q 0, and let Sσ be the subshift generated by it. Then in every word w P Sσ every letter 1 is preceded by 0, and the corresponding subword 01 is the image of a letter 0. The remaining letters 0 are the images of 1, so that the sequence w is uniquely decomposed into a concatenation of σ-images of 1 letters of a sequence w P Sσ.

In fact, the above examples are fairly typical. Namely, we have the following result of Moss´e,see [Ku03, Theorem 4.36]. Theorem 1.3.29. Let σ : X ÝÑ X¦ be a primitive aperiodic substitution. Then it is recognizalbe, i.e., every sequence w in the subshift Fσ generated by σ can be uniquely decomposed into a concatenation of subwords . . . σpy¡1qσpy0qσpy1q ... for a word . . . y¡1y0y1 ... P Fσ.

Here we call a substitution σ : X ÝÑ X¦ is periodic if it generates a finite subshift, i.e., if every element of the subshift generated by σ is a periodic sequence. Otherwise, it is called aperiodic. ¦ Let σ : X ÝÑ X be a primitive substitution generating a subshift Fσ, and let Bσ be the stationary Vershik-Bratteli diagram defined by the constant sequence σ. As described above, every path γ of Bσ defines an infinite sequence w8pγq. Let us assume for a moment that Fσ is infinite, i.e., that σ is aperiodic. Then Fσ is a minimal subshift, hence it can be defined by a properly ordered Vershik-Bratteli diagram. Note that Bσ is not properly ordered in general. Moreover, minimal and maximal paths of Bσ define one- sided sequences, and it is possible that they can be extended to elements of Fσ in many ways, so some paths of Bσ will correspond to several elements of Fσ. Example 1.3.30. The left-hand side part of Figure 1.19 shows the Vershik- Bratteli diagrams associated with the Fibonacci substitution 0 ÞÑ 01, 1 ÞÑ 0 (see Example 1.2.22) Note that the diagram has one minimal path (passing through the vertices 0, 0, 0, 0,...) and two maximal paths (passing through the vertices 1, 0, 1, 0,... and 0, 1, 0, 1,...). The corresponding words w8pγq are the right- infinite limit 01001010 ... of σnp0q, the left-infinite limit ... 01001010 of σ2np0q, and the left-infinite limit ... 01001001 of σ2np1q, respectively. Note 56 1. Dynamical systems

Figure 1.19. Vershik-Bratteli diagrams of Fibonacci and Thue-Morse substitutions that concatenations of the right-infinite word with both left-infinite words belong to the subshift Fσ generated by σ. It follows that the minimal path in the Vershik-Bratteli diagram represents two different points of Fσ.

Proposition 1.3.31. The Vershik-Bratteli diagram Bσ associated with a substitution σ : X ÝÑ X¦ is properly ordered if and only if there exists k ¥ 1 k and letters x0, x1 P X such that for every x P X the words σ pxq starts with x0 and ends with x1.

Proof. Consider the map α : X ÝÑ X mapping x to the first letter of σpxq. Note that αkpxq is equal to the first letter of σkpxq. The orbit of every letter x P X under the iterations of α is eventually periodic, i.e., there exist m, n such that αmpxq αm npxq. Note that if x belongs to a cycle, i.e., if x αnpxq for some n ¥ 1, then there exists a minimal path starting in x and in every letter of the sequence αipxq, i ¥ 1. Since there exists only one minimal path, we must have x αpxq. Similarly, there can be only one point of X belonging to an α-cycle, and it is an α- fixed point. Denote it by x0. Then for every x P X there exists n such that n α pxq x0. Denoting by k the maximal value of such numbers n, we will k get that the first letter of σ pxq is x0 for all x P X. The same argument proves the statement for the last letter. Conversely, if σ satisfies the conditions of the proposition, then it is easy to see that there exist unique minimal (resp. maximal) path corresponding n n to the first (resp. last) letters of the words σ px0q (resp. σ px1q). The same 1.3. Minimal Cantor systems 57 argument shows that there exists k such that the last letters of σkpxq are equal for all x P X. There is a simple algorithm described in [DHS99] producing a properly ordered stationary Vershik-Bratteli diagram B starting from a proper substitution σ, such that the adic transformation on B is topologically conjugate with the subshift generated by σ. Here we extend the notion of a stationary Vershik-Bratteli diagram by allowing the first level to be different from the subsequent levels. Namely, we adopt the following definition. Definition 1.3.32. A stationary Vershik-Bratteli diagram is the diagram associated with an eventually constant sequence of substitutions. ¦ Let σ : X ÝÑ X be a primitive aperiodic substitution, and let Fσ be the subshift generated by σ. After replacing σ by some iterate σn, we may assume that there exists a word ξ . . . x¡2x¡1.x0x1 ... P Fσ such that

. . . x¡2x¡1.x0x1 ... . . . σpx¡2qσpx¡1q.σpx0qσpx1q ....

Since the subshift Fσ is minimal, the subword x¡1x0 appears in ξ infinitely many times with uniformly bounded gaps between consecutive occurrences. More formally, we say that k P Z is an occurrence of x¡1x0 in p q a word yn nPZ if yk¡1 x¡1 and yk x0. Two occurrences k1 k2 are consecutive if there is no occurrence k such that k1 k k2. It follows from Proposition 1.2.4 that there exists a uniform upper bound on k2 ¡ k1 for any consecutive occurrences of x¡1x0 in any element of Fσ. p q If k1 k2 are consecutive occurrences of x¡1x0 in xn nPZ, then we call the word xk1 xk1 1 . . . xk2¡1 the return word for x¡1x0. Let Rx¡1x0 be the set of all return words. It is finite, since the length of the return words is p q P uniformly bounded. By minimality of Fσ, every sequence yn nPZ Fσ can be uniquely decomposed into a concatenation of elements of Rx¡1x0 : just p q cut yn nPZ inside every subword yk¡1yk x¡1x0. P Suppose that w Rx¡1x0 . Then x¡1wx0 belongs to the language of Fσ, the first letter of w is x0, the last letter of w is x¡1. It follows that σpx¡1wx0q σpx¡1qσpwqσpx0q also belongs to the language of Fσ. The first letter of σpwq and σpx0q is x0, the last letter of σpwq and σpx¡1q is x¡1. It follows that if we cut σpwq at every occurrence of x¡1x0, we will p q decompose σ w into a product of elements of Rx¡1x0 . ¦ We have proved that the sub-semigroup of X generated by Rx¡1x0 is σ-invariant. It follows that the restriction of σ to this semigroup is a sub- ÝÑ ¦ stitution, which we will denote by φ : Rx¡1x0 Rx¡1x0 . P Let w0, w1 Rx¡1x0 be the prefix of x0x1 ... and a the suffix of . . . x¡2x¡1, respectively. The words . . . x¡2x¡1 and x0x1 ... are σ-invariants and are lim- np q np q Ñ 8 P its of the words σ x¡1 and σ x0 for n . Since every word w Rx¡1x0 58 1. Dynamical systems

k starts with x0 and ends with x¡1, there exists k such that σ pwq starts with w0 and ends with w1. It follows by Proposition 1.3.31 that the Vershik- Bratteli diagram of φ is properly ordered. Let us add on top of this diagram ÝÑ ¦ one more level associated with the substitution φ0 : Rx¡1x0 X mapping P ¦ every word w Rx¡1x0 to itself (but as an element of X ). We get a stationary Vershik-Bratteli diagram such that its adic transformation is topologically conjugate to Fσ. This diagram is equivalent to the Vershik-Bratteli diagram obtained from the diagram associated with φ, φ, φ, . . . by adding P one vertex on the top level and connecting it to each vertex w Rx¡1x0 by |w| edges. Example 1.3.33. Let us consider the Fibonacci substitution σp0q 01, σp1q 0. Let us pass to the second iterate σ2p0q 010, σ2p1q 01, and consider the corresponding σ2-invariant sequence ... 01001001.01001010 ...

We have R10 tw0 01, w1 001u, and 2 σ p01q 01|001 w0w1 2 σ p001q 01|001|001 w0w1w1. We get the substitution

φ : w0 ÞÑ w0w1, w1 ÞÑ w0w1w1.

It follows that the subshift generated by the Fibonacci substitution is conjugate to the adic transformation shown on the left-hand side part of Figure 1.20. Note that it is equivalent to the diagram shown on the right- hands side part of the same ﬁgure. Both diagrams are properly ordered.

1.3.8. Thue-Morse subshift. The Thue-Morse subshift τ ü T is generated by the substitution σp0q 01, σp1q 10, see Example 1.2.21. The associated Vershik-Bratteli diagram is shown on the right-hand side of Figure 1.19. Note that this diagram has two minimal paths, corresponding to two inﬁnite to the right limits of the words σnp0q and σnp1q, respectively:

w0 0110100110010110 . . . , w1 1001011001101001 .... They are represented on the Vershik-Bratteli diagram by the paths consisting of vertical edges only. 1.3. Minimal Cantor systems 59

Figure 1.20. Vershik-Bratteli diagrams of the Fibonacci subshift

Similarly, it has two maximal paths, coresponding to two infinite to the left limits: ¡ ¡ w0 ... 0110100110010110, w1 ... 1001011001101001. They are represented by the paths consisting of diagonal edges only. Note p q p q p ¡q ¡ p ¡q ¡ that σ w0 w0, σ w1 w1, σ w0 w1 , and σ w1 w0 . For every word w P T there is a unique path in the Vershik-Bratteli diagram producing w (or a one-sided infinite subword of w containing wp0q) as an inductive limit of words σnpxq, x P ta, bu, in the way described in 1.3.7, see Example 1.3.27. ¡ ¡ ¡ ¡ All four concatenations w0 w0, w0 w1, w1 w0, w1 w1 belong to the subshift generated by σ. It follows that each of the maximal and minimal paths in the Vershik-Bratteli diagram represents two points of the subshift. Moreover, the adic transformation from the set of non-maximal paths to the set of non-minimal paths does not admit a continuous extension to the space of all paths. Non-existence of a continuous extension can be easily corrected by “col- laring”, i.e., applying a sliding block map (see Definition 1.2.15) to the substitution. Let us use the block map

. . . x¡1 . x0x1 ... ÞÑ ... px¡2x¡1q . px¡1x0qpx0x1q .... 60 1. Dynamical systems

Figure 1.21. A continuous diagram of the Thue-Morse subshift

We will write the symbol xy of the block code as xy in order to stress that it replaces letter y in the original sequence. Applying σ to two-letter words:

σp00q 01 ¤ 01, σp11q 10 ¤ 10, σp10q 10 ¤ 01, σp01q 01 ¤ 10 we see that the substitution induced on the block code is

σp00q 10 10, σp11q 01 10,

σp10q 00 01, σp01q 11 10.

The Vershik-Bratteli diagram of this substitution is shown on Figure 1.21, where minimal edges are red and maximal edges are black. Note that we have four minimal and four maximal edges, but this time the corresponding one-sided infinite paths are matched to each other in a unique way, since each letter xi of the code remembers the previous letter of the original sequence w P T t0, 1uZ, and the sequence w0, w1 start with different letters. Every path in this diagram corresponds to a unique point of the subshift. Let us show how to construct a properly ordered Vershik-Bratteli diagram for the Thue-Morse subshift, using the return words, as it is described in the previous subsection. Since the substitution σ does not have fixed points on T , we will need to consider its second interation

σ2p0q 0110, σ2p1q 1001. 1.3. Minimal Cantor systems 61

Figure 1.22. A well ordered Bratteli-Vershik diagram of the Thue- Morse subshift

¡ Let us use the sequence w0 .w0 as our ﬁxed point, and consider the set of return words R00. We have

w0 011010|0110|01011010|010110|011010|0110| ...

We get R00 t011010, 0110, 01011010, 010110u. Let us index them in the order of their ﬁrst appearance in wa:

v1 011010, v2 0110, v3 01011010, v4 010110. We have 2 σ pv1q 011010|0110|01011010|010110 v1v2v3v4, 2 σ pv2q 011010|0110|010110 v1v2v4, 2 σ pv3q 011010|01011010|0110|01011010|010110 v1v3v2v3v4, 2 σ pv4q 011010|01011010|0110|010110 v1v3v2v4. The associated Vershik-Bratteli diagram is rather messy to draw. In- 2 stead, let us decompose the substitution φ σ | ¦ into a composition R00 φ φ2 ¥ φ1 of two substitutions

φ1pv1q x1y1, φ1pv2q x1y2,

φ1pv3q x2y1, φ1pv4q x2y2, and

φ2px1q v1v2, φ2px2q v1v3v2,

φ2py1q v3v4, φ2py2q v4. We get then the diagram shown on Figure 1.22 (except for the ﬁrst level...). 62 1. Dynamical systems

Recall that the Vershik-Bratteli diagram of the odometer is the diagram consisting of one vertex and two edges on each level. Let us denote the edges by 0 and 1 with the ordering 0 1. Then the adic transformation will be the usual binary adding machine action. We have a natural semiconjugacy π : T ÝÑ t0, 1uω from the Thue- Morse subshift to the binary odometer. Namely, let w P T , and let γ pe1, e2,...q be the corresponding path in the Vershik-Bratteli diagram of the substitution. Then πpe1, e2,...q x1x2 ..., where xi 0 if ei is minimal, and xi 1 if ei is maximal. It follows directly from the deﬁnition of the adic transformation that π is a semiconjugacy.

Proposition 1.3.34. If the sequence w P t0, 1uω is not eventually constant, then π¡1pwq consists of two elements. Otherwise it consists of four elements.

Proof. It is easy to see from the structure of the Vershik-Bratteli diagram of the substitution σ that for every w P t0, 1uω there are exactly two paths in the diagram (one starting in a and one starting in b) such that every w P T corresponding to these paths is mapped by π to w. The statement about the size of π¡1pwq follows now from our analysis of the relation of the diagram with T .

1.3.9. Vershik-Bratteli diagrams of irrational rotations. As an example of application of Theorem 1.3.20, let us ﬁnd Vershik-Bratteli diagrams realizing the minimal homeomorphisms R˜θ ü Xθ of the Cantor set deﬁned in 1.3.1.2. Let θ P p0, 1q be an irrational number. Consider the continued fraction expansion

1 (1.4) θ . 1 a1 1 a2 1 a 3 . ..

The positive integers ai are found by the following recurrent rule. Set θ1 θ, and then t ¡1u ¡1 ¡ an θn , θn 1 θn an. Note that the last equality is equivalent to θ 1 . n an θn 1 Let us change the recurrent rule by setting t ¡1u ¡ p ¡1 ¡ q (1.5) bn θn , θn 1 1 θn bn , 1.3. Minimal Cantor systems 63 so that we have θ 1 , and n bn 1¡θn 1

1 θ . ¡ 1 b1 1 ¡ 1 b2 1 ¡ 1 b 1 3 . .. The expansions (1.5) are called negative-regular continued fractions. ¡1 ¡ We could also make diﬀerent choices between θn 1 θn an and ¡ p ¡1 ¡ q θn 1 1 θn an . Such generalized continued fractions are called semi-regular continued fractions, see [Per54, Chapter V], where they are called halbregelm¨aßige. An algorithm transforming semi-regular continued fractions to the classical expansion (1.4) is described in [Per54, V.40]. In the particular case of the sequence pbnq from (1.5) we get the following.

Proposition 1.3.35. The sequence pb1, b2,...q is equal to

a1, 1loooomoooon, 1,..., 1, a3 1, loooomoooon1, 1,..., 1, a5 1, loooomoooon1, 1,..., 1, a7 1,....

a2 ¡ 1 times a4 ¡ 1 times a6 ¡ 1 times

In particular, inﬁnitely many entries of the sequence bn greater than 1.

Note that the negative-regular continued fraction (1.5) with the constant sequence bn 1 is equal to 1.

1 1 Proof. If ξ, then 1 ξ . Conse- 1 ξ 1 1 1 1 ξ a3 a3 1 1 1 a a 4 . 4 . .. .. quently, the proposition will follow from the identity

1 1 (1.6) , 1 1 a1 a1 1 ¡ a2 ξ 1 2 ¡ 1 2 ¡ ... 1 2 ¡ 1 1 ξ where 2 appears a2 ¡ 1 times on the right-hand side. 64 1. Dynamical systems

It is easy to prove by induction on n that 1 n ¡ pn ¡ 1qx , 1 pn 1q ¡ nx 2 ¡ 1 2 ¡ ... 2 ¡ x where 2 appears n times on the left-hand side. Consequently, if x 1 , the right-hand side of (1.6) is equal to 1 1 ξ 1 1 ¡ . a2¡1¡pa2¡2qx 1 x a1 1 ¡ a1 ¡p ¡ q a2¡pa2¡1qx a2 a2 1 x

Substitution x ξ into 1¡x gives ξ 1 a2¡pa2¡1qx ¡ ξ 1 1 1 ξ 1 , ¡ p ¡ q ξ a pξ 1q ¡ pa ¡ 1qξ a ξ a2 a2 1 ξ 1 2 2 2 which ﬁnishes the proof.

Theorem 1.3.36. Let θ P p0, 1q be an irrational number, and let pb1, b2,...q be the sequences of positive integers such that 1 θ . 1 b1 1 ¡ 1 b2 1 ¡ 1 b 1 ¡ 3 . .. Deﬁne the substitutions k k¡1 ψkp0q 0 1, ψkp1q 0 1.

Then the dynamical system R˜θ ü Xθ is topologically conjugate to the substitutional system generated by the sequence

ψb1 , ψb2 , ψb3 ,... and to the adic transformation on the associated Vershik-Bratteli diagram.

The Vershik-Bratteli diagram of the substitution ψ3 is shown on the left-hand side of Figure 1.23.

Proof. Let R1 be the partition r0, 1¡θs, r1¡θ, 1s of the Cantor set Xθ seen as Rokhlin-Kakutani partitions with base equal to the whole space Xθ. (We will drop 0 and ¡0 from the interval notation, since it will be always clear what are the intervals.) 1.3. Minimal Cantor systems 65

Figure 1.23. Vershik-Bratteli diagrams of a rotation

Figure 1.24. Rokhlin-Kakutani partition of a rotation

Let us construct a Rokhlin-KakutaniX \ partition subordinate to R1 with r s 1 P r ¡ s the base 0, θ . Let a b1 θ . If x 0, 1 aθ , then the smallest positive integer n such that Rnpxq P r0, θs is equal to a 1, while for x P r1¡aθ, θs it is a. The first return map maps r0, 1¡aθs to rpa 1qθ¡1, 1¡aθ pa 1qθ¡1s rpa 1qθ ¡ 1, θs, and maps r1 ¡ aθ, θs to r0, pa 1qθ ¡ 1s. Hence the first return map swaps the intervals r0, 1¡aθs and r1¡aθ, θs. We define therefore R2 as the Rokhlin-Kakutani partition with the base r0, θs and two towers:

2 a Tr0,1¡aθs tr0, 1 ¡ aθs,Rpr0, 1 ¡ aθsq,R pr0, 1 ¡ aθs,...R pr0, 1 ¡ aθsqu and

a¡1 Tr1¡aθ,θs tr1 ¡ aθ, θs,Rpr1 ¡ aθ, θsq,...,R pr1 ¡ aθ, θsqu, see Figure 1.24, where the intervals of the ﬁrst tower are black, and the intervals from the second tower are red.

The interval r0, 1 ¡ θs P R1 is equal to the union of the following 2a ¡ 1 elements of R2:

r0, 1 ¡ aθs Y Rpr0, 1 ¡ aθsq Y ¤ ¤ ¤ Y Ra¡1pr0, 1 ¡ aθsqY r1 ¡ aθ, θs Y Rpr1 ¡ aθ, θsq Y ¤ ¤ ¤ Ra¡2pr1 ¡ aθ, θsq r0, 1 ¡ aθs Y rθ, 1 ¡ pa ¡ 1qθs Y ¤ ¤ ¤ Y rpa ¡ 1qθ, 1 ¡ θsY r1 ¡ aθ, θs Y r1 ¡ pa ¡ 1qθ, 2θs Y ¤ ¤ ¤ r1 ¡ 2θ, pa ¡ 1qθs.

The interval r1¡θ, 1s P R1 is equal to the union of the following two elements of R2: Ra¡1pr1 ¡ aθ, θsq Y Rapr0, 1 ¡ aθsq r1 ¡ θ, aθs Y raθ, 1s. 66 1. Dynamical systems

In other words, r1 ¡ θ, 1s is the union of the two last elements of the towers, while r0, 1 ¡ θs is the union of all the remaining elements. See Figure 1.24 where these decompositions are shown. We see that the level of the Vershik-Bratteli diagram associated with the pair R1 and R2 has a edges Tr0,1¡aθs P V2 to tr0, 1 ¡ θsu P V1, one edge connecting Tr0,1¡aθs to tr1 ¡ θ, 1su P V1, a ¡ 1 edges connecting Tr1¡aθ,θs P V2 to tr0, 1 ¡ θsu, and one edge connecting Tr1¡aθ,θs to tr1 ¡ θ, 1su. The edge connecting a vertex of V2 to tr1¡θ, 1su is greater (comes later in the ordering of the diagram) than the edges connecting it to tr0, 1 ¡ θsu. In other words, it is the diagram of the substitution a a¡1 ψa : 0 ÞÑ 0 1, 1 ÞÑ 0 1, if we label the towers r0, 1 ¡ θs and Tr0,1¡aθs by 0, and the towers r1 ¡ θ, 1s and Tr1¡aθ,θs by 1. We can identify the base r0, θs of R2 with the interval r0, 1s by the transformation x ÞÑ x{θ. Then the partition r0, 1 ¡ aθs, r1 ¡ aθ, θs is transformed to the partition r0, θ¡1 ¡as, rθ¡1 ¡a, 1s, and the ﬁrst return map swaps these intervals, i.e., is a rotation by θ1 1 ¡ pθ¡1 ¡ aq. (Note that if we identify the base r0, θs with the interval r0, 1s by the transformation x ÞÑ 1 ¡ x{θ, then the ﬁrst return map is a rotation by θ1 θ¡1 ¡ a.) We apply our procedure again to the rotation by θ1. We will get a sequence Rn of Rokhlin-Kakutani partitions such that its Vershik-Bratteli diagram is the diagram associated with the sequence

ψb1 , ψb2 , ψb3 ,....

The bases of the partitions pRnq are the segments r0, θns, where θn are 1 deﬁned by the condition θn 1 . Consequently, the intersection of an 1¡ θn 1 the bases consists of 0 only. As the length of the towers grow to inﬁnity, for r ¡ s ˜k every m there exists n such that images of 0, 1 anθn under Rθ belong to Rn for all k 0, 1, . . . , m. It follows that any two points t ¡ 0, t 0, where t is in the forward orbit of 0 under Rθ are separated by some partition Rn.

The union of the top elements of the towers of Rn is the interval r1 ¡ θn, 1s, and intersection of these intervals is t1u. By the same argument as above, this shows that any two points t ¡ 0, t 0, where t is in the backward orbit of 0 under Rθ are separated by some partition Rn.

It follows that adic transformation of the Vershi-Bratteli diagram of pRnq ˜ is topologically conjugate to Rθ.

Deﬁnition 1.3.37. We say that a subshift F XZ is Sturmian if pF pnq n 1 for all n ¥ 0, where pF is the complexity function, see... 1.3. Minimal Cantor systems 67

Note that since pF p1q 2, we may assume that |X| 2.

Theorem 1.3.38 (Hedlund-Morse). Suppose that a subshift F t0, 1uZ is minimal and satisﬁes pF pnq n 1 for all n ¥ 1. Then there exists an irrational number θ such that F Fθ.

Reference from Hedlund-Morse... For the countable case, see...

Proof. Denote by Wn the set of words of length n belonging to WF . Con- sider the map vx ÞÑ v from the Wn 1 to Wn. It is surjective, and since |Wn 1| n 2 and |Wn| n 1, there exists one word wn P Wn (which we will call special) that has two preimages. All the other words v P Wn have one preimage. Conversely, if for every n ¥ 1 there exists exactly one word wn P Wn with two preimages and all the other words have only one preimage, then |Wn 1| |Wn| 1, and |Wn| n 1 for all n.

If wn P WF is the special word, i.e., if wn0, wn1 P WF , then every its suffix of wn is also special. Since there is only one special word of every length, it follows that special words are suffixes of one left-infinite word ¡ω w8 . . . x2x1x0 P X . We will call it the left-infinite special word. k Let k ¥ 0 be the smallest number such that 10 1 belongs to WF . If the k 1 word 0 does not belong to WF , then there are exactly k letters 0 between any two consecutive 1s. But then F is finite. Note also that no element of F contains an infinite string of 0s, since this would contradict minimality. k 1 k k k 1 It follows that the words 0 10 and 0 10 belong to WF . Consequently, i k 1¡i all the words of the form 0 10 for i 0, 1, . . . , k 1 belong to WF . We k also have 10 1 P WF . This is already a list of k 3 words in WF of length k 2. It follows that there are no other words of length k 2 in WF . In k 2 particular 0 R WF , i.e., the number of zeros between any two consecutive ones is either k or k 1. Consequently, every element of F can be written as a concatenation of the words 0k1 and 0k 11. Consider now the subshift F 1 over the alphabet t0k1, 0k 11u consisting of sequences obtained by factoring sequences w P F into concatenations of subwords 10k and 10k 1. Let us show that the obtained subshift F 1 is also Sturmian. It is enough to show that for every n there exists a unique word v P WF 1 of length n such k k 1 1 that v ¤ p0 1q, v ¤ p0 1q P F . Let us show at first the existence. Let w8 be the left-infinite special word for F. Then w80 and w81 are subwords of k elements of F. It follows that 10 is a suffix of w8, and w8 can be factored k k k 1 into a concatenation . . . a3a20 , where ai P t0 1, 0 1u. But then both k k 1 1 . . . a3a20 1 and . . . a3a20 1 are subwords of elements of F , hence . . . a3a2 68 1. Dynamical systems is a left-infinite special word of F 1. For every n the suffix of length n of this word will be special for F 1. 1 In order to prove the uniqueness, it is enough to show that if v1, v2 P F k k 1 k k 1 are such that tv1p0 1q, v1p0 1q, v2p0 1q, v2p0 1qu WF 1 , then one of the k 1 words v1, v2 is a suffix of the other. Note that vip0 1q P WF 1 implies that k 1 k k 1 k k 1 vi0 P WF , so that we have that tv10 1, v110 , v20 1, v20 u WF , k k i.e., that v10 and v20 are special words for F. But we know that all special words of F are suffixes of the unique left-infinite special word. It follows that one of the words v1, v2 is a suffix of the other. We see that F 1 is also a Sturmian minimal subshift (minimality of F 1 follows directly from the minimality of F by Proposition 1.2.4). Let us relabel the letters of the alphabet t0k 11, 0k1u by the letters 0, 1 using the substitution k 1 k ψk 1 : 0 ÞÑ 0 1, 1 ÞÑ 0 1. Note that this is the same substitution as in Theorem 1.3.36. We will identify then F 1 with a Sturmian subshift of t0, 1uZ. Continue the above construction with F replaced by the new F 1 t0, 1uZ. We will get a sequence of positive integers k1, k2,..., a sequence of subshifts Fn, and a sequence of substitutions

ÞÑ kn 1 ÞÑ kn ψkn 1 : 0 0 1, 1 0 1.

The elements of Fn are obtained from the elements of Fn 1 by applying ψkn (and then taking all shifts). Let B be the Vershik-Bratteli diagram associated with the obtained sequence ψkn 1, and let us show that every path in B corresponds to exactly one element of F. It is enough to check the minimal and the maximal paths.

But ψ1 is the substitution 0 ÞÑ 01, 1 ÞÑ 1 generating the subshift consisting of the constant 1 sequence. Consequently, the sequence kn is not eventually equal to zero, since then F would be ﬁnite. It follows that the only minimal path is the unique path passing through the vertices 0 P Vn. The corresponding right-inﬁnite sequence is limit of the

¥ ¥¤ ¤ ¤ p q kn 1p q kn 1 words ψk1 1 ψk2 1 ψkn 1 0 . Note ψ 0 that it begins with 0 1, kn 1 which has a unique allowed extension to the left 10 1, since kn 1 is the maximal allowed number of zeros between two consecutive ones in Fn. It ¥ ¥ ¤ ¤ ¤ p q follows that in every element of F the word ψk1 1 ψk2 1 ψkn 1 0 is ¥ ¥ ¤ ¤ ¤ p q preceded by ψk1 1 ψk2 1 ψkn¡1 1 1 . The length of this word goes to inﬁnity, since kn is not eventually zero. Consequently, the right-inﬁnite sequence corresponding to the minimal path has a unique extension to an element of F. Let us show that the maximal path corresponds to a unique element of F. It is enough to prove this statement for any Fn. It follows that we may 1.3. Minimal Cantor systems 69

assume that k1 0. The only maximal path is the path passing through the vertices 1 P Vn. The corresponding left-infinite sequence is the left-infinite ¥ ¥¤ ¤ ¤ p q limit of the words ψk1 1 ψk2 1 ψkn 1 1 . Then 1 is the last letter of this word, and then the only possible one-letter continuation is 10, as k1 ¡ 0. The proof is finished in the same way as for the case of the minimal path. This shows that the diagram B models the subshift F. Theorem 1.3.36 shows (together with its proof, as we have to check that the encodings by 0 and 1 agree) that F coincides with the system R˜θ ü Xθ for 1 θ . 1 k1 2 ¡ 1 k2 2 ¡ 1 k 2 ¡ 3 . .. We want to consider now the classical continued fractions... Let k 1 k ψk 1 : 0 ÞÑ 0 1, 1 ÞÑ 0 1, be the substitution from Theorem 1.3.36. Define also the substitutions k k 1 φk 1 : 0 ÞÑ 0 1, 1 ÞÑ 0 1. Proposition 1.3.39. The subshift generated by the sequence

φk1 1, φk2 1, φk3 1,... coincides with the subshift generated by the sequence

k2 k4 k6 ψk1 1, ψ1 , ψk3 2, ψ1 , ψk5 2, ψ1 , ψk7 2,... See Deﬁnition 1.3.26 of subshifts generated by sequences of substitutions.

Proof. Denote η : 0 ÞÑ 0, 1 ÞÑ 01. ¥ ¥ ¥ k2 ¥ Then ψk 2 η ψk 1. Let us compare now φk1 1 φk2 1 with ψk1 1 ψ1 η. We have

¥ p q p k2 q p k1 qk2 k1 1 φk1 1 φk2 1 0 φk1 1 0 1 0 1 0 1 and

¥ k2 ¥ p q p k2 q k1 1 p k1 qk2 ψk1 1 ψ1 η 0 ψk1 1 01 0 1 0 1 . We have

¥ p q p k1 1 q p k1 qk1 1 k1 1 φk1 1 φk2 1 1 φk1 1 0 1 0 1 0 1 and

¥ k2 ¥ p q ¥ k2 p q p k2 1q k1 1 p k1 qk2 1 ψk1 1 ψ η 1 ψk1 1 ψ 01 ψk1 1 01 0 1 0 1 . 70 1. Dynamical systems

¥ It follows that application of φk1 1 φk2 1 to a bi-inﬁnite word produces,

¥ k2 ¥ p q k1 1 up to a shift, the same word as application of ψk1 1 ψ η u 0 v. It follows that the subshift generated by the sequence

φk1 1, φk2 1, φk3 1,... is the same as the subshift generated by the sequence

k2 k4 ψk1 1, ψ , η, ψk3 1, ψ , η, . . . . The last sequence is equivalent to

k2 k4 ψk1 1, ψ , ψk3 2, ψ , ψk5 2,....

Theorem 1.3.40. Let θ P p0, 1q be an irrational number, and let pa1, a2,...q be the sequence of positive integers such that 1 θ . 1 a1 1 a2 1 a 3 . .. Deﬁne the substitutions k¡1 k φkp0q 0 1, φkp1q 0 1.

Then the dynamical system R˜θ ü Xθ is topologically conjugate to the system generated by the sequence

φa1 , φa2 , φa3 ,....

If inﬁnitely many of the values of ai are greater than 1, then it is also topologically conjugate to the adic transformation of the Vershiki-Bratteli diagram associated with the above sequence. ? 1 5 Example 1.3.41. Let ϕ? 2 be the golden mean, and consider the rotation by ϕ ϕ ¡ 1 ¡1 5 pmod 1q. 2 ? The inverse rotation is by θ 1 ¡ ϕ 3¡ 5 . We have ? 2 ? 2 6 2 5 3 5 θ¡1 ? , 3 ¡ 5 4 2 hence a1 2, and ? ? 3 5 3 ¡ 5 θ 3 ¡ θ. 2 2 2 1.3. Minimal Cantor systems 71

Figure 1.25. Two Vershik-Bratteli diagrams of the golden mean rotation

It follows that 1 θ 1 ¡ ϕ 1 2 1 ¡ 1 2 1 ¡ 1 2 1 ¡ ... It is also well known that 1 ϕ 1 1 1 1 1 1 ... Consequently, the rotation by ϕ can be described by the Vershik-Bratteli diagrams shown on Figure 1.25. The left-hand side part is obtained by changing to the opposite the ordering of the edges of the diagram associated with the sequence pRnq for the rotation by θ 1 ¡ ϕ, while the right-hands p 1 q side part shows the diagram associated with the sequence Rn for ϕ. We have highlighted the minimal edges. Note that there are two minimal paths in the right-hand side diagram.

1.3.10. Linearly repetitive systems. By Proposition 1.2.4, if pF, sq is a minimal subshift, then for every ﬁnite word v P WF there exists Nv such that for every w P F there exists 0 ¤ k ¤ Nv ¡ 1 such that the subword 72 1. Dynamical systems

1 1 1 1 k w p1qw p2q . . . w p|v|q of w s pwq is equal to v. Denote by RF pnq the maximum of Nv for all words v of length at most n. Deﬁnition 1.3.42. We say that a subshift F is linearly repetitive if there exists C ¡ 1 such that RF pnq ¤ Cn for all n ¥ 1.

The following is straightforward.

Proposition 1.3.43. Suppose that F 1 pXkqZ is the image of a minimal subshift F XZ under a sliding block map. Then

RF 1 pnq RF pn kq for all n ¥ 1. In particular, F is linearly repetitive if and only if so is F 1.

Theorem 1.3.44. Minimal substitutional subshifts are linearly repetitive.

Proof. It is shown in the proof of Proposition 1.2.27 that the image of every minimal substitutional subshift under some sliding block map is generated by a primitive substitution. It follows then from Proposition 1.3.43 that it is enough to consider only the case of a subshift F generated by a primitive substitution σ : X ÝÑ X¦. It was shown in the proof of Theorem 1.2.34 that there exists a constant C, depending only on F, such that for every n ¡ 1 there exists k such that k for every x P X we have n ¤ |σ pxq| ¤ Cn. Then every word v P WF of k length n is a subword of a word of the form σ pxyq for xy P WF . It follows p q ¤ p q ¤ that RF n RF 2 Cn.

See another proof in [DL06, Theorem 1], where this theorem was proved for the ﬁrst time. More generally, we have the following characterization of linearly repetitive subshifts, see [Dur03].

Theorem 1.3.45. A subshift is linearly repetitive if and only if it is topologically conjugate to the subshift generated by a sequence of substitutions

¦ ÐÝσ1 ¦ ÐÝσ2 ¦ ÐÝσ3 ¤ ¤ ¤ X1 X2 X3 such that

(1) the set tσn : n 1, 2,...u is ﬁnite;

(2) there exists s such that for every x P Xn and y P Xn s 1 the letter x appears in the word σn ¥ σn 1 ¥ ¤ ¤ ¤ ¥ σn spyq;

(3) there exist an, bn P Xn such that for every x P Xn 1 the word σnpxq begins with an and ends with bn. 1.3. Minimal Cantor systems 73

Denote by pF pnq the complexity of the subshift F, i.e., the number of subwords of length n in elements of F. We have an obvious estimate pF pnq ¤ RF pnq, since every word v P WF of length n appears as a subword of every the word u P WF of length n RF pnq¡1. On the other hand, RF pnq can grow much faster than pF pnq. For example, RF can grow arbitrarily fast even for Sturmian sequences. Namely, we have the following theorem, see ...

Theorem 1.3.46. Let Xθ t0, 1uZ be the Sturmian shift deﬁned by an irrational number θ P p0, 1q, as in Proposition 1.2.37, and let a1, a2,... be the terms of the continued fraction expansion of θ (the partial quotients). Then Xθ is linearly repetitive if and only if an is bounded (i.e., if θ is of bounded type).

Proof.¢ (Sketch) Let φai be as in Theorem 1.3.40. Consider the matrix b00 b01 ¥ Bn , where bxy is the number of letters x in the word φa1 b10 b11 ¢ 1 0 φ ¥ ¤ ¤ ¤ ¥ φ pyq. We have B . It follows from the deﬁnition a2 an 0 0 1 of the morphism φai that ¢ an 1 ¡ 1 an 1 B B ¤ . n 1 n 1 1 Consequently, ¢ ¢ ¢ a ¡ 1 a a ¡ 1 a a ¡ 1 a B 1 1 2 2 ¤ ¤ ¤ n n . n 1 1 1 1 1 1

Denote by pn, qn the numerator and the denominator of the fraction 1 , respectively. We have then 1 a1 1 a2 . 1 .. an

pn 1 an 1pn pn¡1, qn 1 an 1qn qn¡1. It is easy to check by induction that then ¢ qn ¡ pn qn ¡ pn qn¡1 ¡ pn¡1 Bn . pn pn pn¡1 ¥ ¥ ¤ ¤ ¤ ¥ p q ¥ ¥ ¤ ¤ ¤ ¥ p q In particular, the lengths of φa1 φa2 φan 0 and φa1 φa2 φan 1 are qn and qn qn¡1, respectively. 74 1. Dynamical systems

Let r qn . Then r a , and we have n qn¡1 1 1

an 1qn qn¡1 1 rn 1 an 1 , qn rn so that 1 rn an , 1 an¡1 1 an¡2 . 1 .. a1 which implies an ¤ rn ¤ an 1. We have |φ ¥ φ ¥ ¤ ¤ ¤ ¥ φ p1q| q q ¡ 1 a1 a2 an n n 1 | ¥ ¥ ¤ ¤ ¤ ¥ p q| 1 φa1 φa2 φan 0 qn rn and ||φ ¥ φ ¥ ¤ ¤ ¤ ¥ φ p0q| q a1 a2 an n || ¥ ¥ ¤ ¤ ¤ ¥ p q| rn. φa1 φa2 φan¡1 0 qn¡1 It follows that these ratios are uniformly bounded away from zero and in- ﬁnity if and only if an is bounded from above. If this is the case, then the proofs Theorems 1.2.34 and 1.3.44 can be repeated for Xθ, which will show the “if” direction. For the “only if” direction it is enough to notice that the word between ¥ ¥ ¤ ¤ ¤ ¥ p q ¥ ¥ two neighboring occurrences of φa1 φa2 φan 1 is either φa1 φa2 ¡ ¤ ¤ ¤ ¥ p an 1 1q ¥ ¥ ¤ ¤ ¤ ¥ p an 1 q φan 0 or φa1 φa2 φan 0 , which shows that p q ¥ RXθ qn qn¡1 qn qn¡1 an 1qn qn qn 1. We have ¡1 qn qn 1 1 rn 1 1 an 1 rn ¥ an 1 ¡1 ¡1 1 , qn qn¡1 1 rn 1 rn 2 as rn ¥ 1 and the expression is increasing with rn. It follows that if the R pnq Xθ sequence an is not bounded from above, then so is n .

1.4. Hyperbolic dynamics

General discussion.. Expanding maps are very well understood and have rich analytic and algebraic structure. See, for instance [Haisinki-Pilgrim...] and... One of the main subjects of Chapter 5 is the algebraic theory of expanding maps, where we will establish a functorial bijection between expanding covering maps and a class of groups (more precisely groups bisets...)... The case of hyperbolic homeomorphisms (Ruelle-Smale spaces) is much less understood, and many questions that are relatively easy in the expanding case are wide open in the case of homeomorphisms.... 1.4. Hyperbolic dynamics 75

1.4.1. Expanding maps: definitions and examples. Definition 1.4.1. Let X be a compact space. A map f ü X is expanding if it generates an expansive action of the semigroup N, i.e., if there exists a neighborhood U X ¢ X of the diagonal such that pf kpxq, f kpyqq P U for all k ¥ 0 implies x y. Definition 1.4.2. We say that a map f ü X on a compact metric space is metrically expanding if there exist ¡ 0, L ¡ 1 such that if x, y P X are such that dpx, yq then dpfpxq, fpyqq ¥ Ldpx, yq.

It is obvious that every metrically expanding map is expanding.

Example 1.4.3. Consider the circle R{Z and the self-covering f ü R{Z given by fpxq kx pmod 1q, for an integer k, |k| ¡ 1. It is metrically expanding, for example with L |k| and 1{|k|.

It is easier in some cases to show that an iteration of a map is metrically expanding. A simple change of the metric shows that then the map itself expanding. Lemma 1.4.4. Suppose that f ü X is a map on a metric space pX , dq such that f n ü X is metrically expanding for some n ¥ 1. Then there exists a metric d1 on X such that f ü X is metrically expanding with respect to d1.

Proof. Suppose that f n is expanding with respect to a metric d. Let and L be as in Deﬁnition 1.4.2 for f n. Consider the metric n¸¡1 d1px, yq L¡k{ndpf kpxq, f kpyqq. k0 Then ¸n d1pfpxq, fpyqq L¡pk¡1q{ndpf kpxq, f kpyqq k1 n¸¡1 L¡pn¡1q{ndpf npxq, f npyqq L1{n L¡k{ndpf kpxq, f kpyqq ¥ k1 n¸¡1 L¡1 1{nLdpx, yq L1{n L¡k{ndpf kpxq, f kpyqq L1{nd1px, yq. k1 1 It follows that f is expanding with respect to d . Example 1.4.5. An endomorphism f ü M of a Riemannian manifold is called expanding if there exist C ¡ 0 and L ¡ 1 such that }Df n~v} ¥ CLn}~v} 76 1. Dynamical systems for every tangent vector ~v. It follows from Lemma 1.4.4 that every expanding endomorphism of a compact Riemannian manifold is an expanding self-covering. Expanding endomorphisms of Riemannian manifolds were studied by M. Shub in [Shu69, Shu70, Hir70]. One of applications of M. Gromov’s theorem on groups of polynomial growth is showing that all expanding endomorphisms of compact Riemannian manifolds are generalizations of Ex- amle 1.4.3, see [Gro81]. Namely, they are all topologically conjugate to endomorphisms of infra-nil-manifolds. Here a manifold M is an infra-nil- manifold if there exists a nilpotent connected Lie group L and a subgroup G L o AutpLq such that G acts freely and properly on L, and M is diffeomorphic to L{G. If F ü L is an expanding automorphism such that F GF ¡1 G, then F induces an expanding endomorphism of L{G M. We will revisit this result in ... Example 1.4.6. We get many more examples of expanding self-coverings f ü X , if we do not require X to be a manifold. A big class of examples is provided by holomorphic dynamics. Namely, every hyperbolic complex rational function is expanding on its Julia set, see 1.5.3. Example 1.4.7. Let s ü XN be the one-sided shift. Consider the metric ¡n dpw1, w2q 2 , where n is the largest non-negative integer such that the beginnings of length n of w1 and w2 coincide. Then s is expanding for 1{2 and L 2. In particular, every one-sided subshift F XN is expanding. This shows that the class of all expanding maps is very big (for example, there exists uncountably many conjugacy classes of one-sided shifts). On the other hand, we will see later that the class of expanding self-coverings is much more rigid. In particular, it contains only countably many topological conjugacy classes, see...

1.4.2. Natural log-scale. Let f ü X be a map, where X is compact. If U X ¢ X is an expansion entourage for f ü X , then U ¡ tpx, yq : py, xq P Uu is also an expansion entourage. Then U ¡ X U is a symmetric expansion entourage. It follows that we may assume without loss of gener- ality that expansion entourages are symmetric. We will also assume that they are closed. Suppose that U is an expansion entourage for a map f ü X . Denote £n ¡k Un f pUq. k0 k k In other words, Un is the set pairs of points px, yq such that pf pxq, f pyqq P ¢ U for all k 0, 1, . . . , n. In particular, U0 U. Denote U¡1 X X . By the deﬁnition of an expansion entourage, n¥0 Un is equal to the diagonal. 1.4. Hyperbolic dynamics 77

The following is a particular case of Lemma 1.2.7.

Lemma 1.4.8. For every neighborhood V of the diagonal, there exists n such Un V .

For subsets A, B of X ¢ X , denote by A ¥ B the set of pairs px, yq such that there exists z such that px, zq P A and pz, yq P B. Note that A ¥ B is the image of the closed subset D tppx1, y1q, px2, y2qq : y1 x2u of A ¢ B under the map ppx1, y1q, px2, y2qq ÞÑ px1, y2q. If A and B are compact, then D is a closed subset of a compact space A ¢ B, hence D is compact, which implies that A ¥ B is compact.

Lemma 1.4.9. There exists ∆ P N such that Un ∆ ¥ Un ∆ Un for all n ¥ 1.

Proof. Suppose that there is no ∆ such that U∆ ¥ U∆ IntpUq. Denote 2 Bk pX r IntpUqq X pUk ¥ Ukq, for k ¥ 0. Then the sets Bk are closed 2 non-empty, and Bk 1 Bk. It follows from compactness of X that the p q p q P intersection k¥1 Bk is non-empty. Let x, y be such that x, y Bk for all k. Let Zk X be the set of points z such that px, zq P Uk and pz, yq P Uk. Since Uk is closed, the set Zk is closed. It is non-empty, by the choice of p q x, y . We also have Zk 1 Zk. It follows that the intersection of all Zk P p q P is non-empty. Let z0 k¥1 Zk. Then x, z0 Uk for all k, hence x z0, and pz0, yq P Uk for all k, hence z0 y, which implies x y, which is a contradiction.

We have shown that there exists ∆ such that U∆ ¥ U∆ U. If px, yq P U∆ n ¥U∆ n, then there exists z such that px, zq P U∆ n and pz, yq P U∆ n. i i i i Then pf pxq, f pzqq P U∆ n¡i U∆ and pf pzq, f pyqq P U∆ n¡i U∆ for i i all i 0, 1, . . . , n. It follows that pf pxq, f pyqq P U∆ ¥ U∆ U, hence p q P ¥ ¥ x, y Un. We have shown that Un ∆ Un ∆ Un for all n 0. Deﬁnition 1.4.10. Denote, for px, yq P X 2, by `px, yq the maximal value of n such that px, yq P Un, and 8 if x y.

Lemma 1.4.9 is reformulated then as follows.

Proposition 1.4.11. There exists ∆ ¡ 0 such that `px, yq ¥ minp`px, zq, `pz, yqq ¡ ∆ for all x, y, z P X .

The function `px, yq measures “closeness” of points of X . The closer points x and y are, the bigger is the value of `px, yq. We will transform this function into a metric in the next subsection. 78 1. Dynamical systems

1.4.3. Log-scales and associated metrics. Generalizing Propositin 1.4.11, we adopt the following deﬁnition.

Deﬁnition 1.4.12. A log-scale on a set X is a map ` : X ¢ X Ñ R Y t8u satisfying the following conditions. (1) `px, yq `py, xq for all x, y P X; (2) `px, yq 8 if and only if x y; (3) there exists ∆ ¡ 0 such that `px, yq ¥ minp`px, zq, `pz, yqq ¡ ∆ for all x, y, z P X.

We say that a metric d on X is associated with the log-scale ` if there exist constants α ¡ 0, c ¡ 1, such that c¡1e¡α`px,yq ¤ dpx, yq ¤ ce¡α`px,yq. The number α is called the exponent of the metric. It is easy to check that if d is any metric on X, then `px, yq ¡ log dpx, yq is a log-scale such that d is associated with `. With this connection between log-scales and metrics in mind, we give the following deﬁnition.

Deﬁnition 1.4.13. We say that two log-scales `1, `2 on X are bi-Lipschitz equivalent if sup |`1px, yq ¡ `2px, yq| 8. x,yPX,xy

Theorem 1.4.14. Let ` be a log-scale on a set X. Then there exists αc P p0, 8s such that for every α P p0, αcq there exists a metric d on X of exponent α associated with `, and for every α ¡ αc such a metric does not exist.

Proof. Consider, for every n P N the graph Γn with the set of vertices X in which two points x, y are connected by an edge if `px, yq ¥ n. Let dn be the combinatorial distance between the vertices of Γn. Lemma 1.4.15. There exists α ¡ 0 and C ¡ 0 such that αpn¡`px,yqq dnpx, yq ¥ Ce for all x, y P X and n P N. Proof. Let ∆ be as in Proposition 1.4.11, and let us prove the lemma for ln 2 p q ¥ ¡ α ∆ . If x0, x1, x2 is a path in Γn, then ` x0, x2 n ∆, hence x0, x2 is p q ¤ 1 p p q q a path in Γn¡∆. It follows that dn¡∆ x, y 2 dn x, y 1 , or

dn ∆px, yq ¥ 2dnpx, yq ¡ 1.

If `px, yq m, then dm 1px, yq ¥ 2, and hence t dm 1 t∆px, yq ¥ 2 1. 1.4. Hyperbolic dynamics 79

Y ] n¡`px,yq¡1 ¡ n¡`px,yq¡1 ¡ It follows that for every n and t ∆ ∆ 1 we have

t pn¡`px,yq¡1¡∆q{∆ αpn¡`px,yqq dnpx, yq ¡ 2 ¡ 2 Ce , p¡1¡∆q{∆ ln 2 where C 2 and α ∆ .

We say that α ¡ 0 is a lower exponent if there exists C ¡ 0 such that α and C satisfy the conditions of Lemma 1.4.15. If α is a lower exponent, then all numbers in the interval p0, αq are lower exponents. Hence, the set of lower exponents is either an interval p0, αcq (including the case αc 8) or an interval p0, αcs. The number αc is called the critical lower exponent, and we are going to prove the theorem for this value. It is easy to see that if α is such that there exists a metric of exponent α, then α is a lower exponent. Let α be a lower exponent, and let β P p0, αq. Let us show that there p q exists° a metric d of exponent β. Denote by dβ x, y the inﬁmum of the ¡ p q m β` xi,xi 1 sum i1 e over all sequences x0 x, x1, x2, . . . , xm y. The function dβ obviously satisﬁes the triangle inequality, dβpx, yq dβpy, xq, ¡β`px,yq and dβpx, yq ¤ e for all x, y P X . It remains to show that there exists ¡β`px,yq C ¡ 0 such that Ce ¤ dβpx, yq. In other words, there exists C is such that ¸m ¡ p q ¡ p q (1.7) Ce β` x,y ¤ e β` xi,xi 1 i1 for all sequences x0, x1, . . . , xm such that x x0 and y xm. ¡αpn¡`px,yqq Let C0 P p0, 1q be such that dnpx, yq ¥ C0e ¡ for all ©x, y P X p ¡ q P β ln C0 2α∆ and n N. Let us prove inequality (1.7) for C exp α¡β .

Lemma 1.4.16. Let x0, x1, . . . , xm be a sequence such that `pxi, xi 1q ¥ n for all i 0, 1, . . . , m ¡ 1. Let n0 ¤ n. Then there exists a sub-sequence p qm y0 x0, y1, . . . , yt¡1, yt xm of the sequence xi i0 such that

n0 ¡ 2∆ ¤ `pyi, yi 1q n0 for all i 0, 1, . . . , t ¡ 1.

Proof. Let us construct the subsequence yi by the following algorithm. Deﬁne y0 x0. Suppose we have deﬁned yi xr for r m. Let s be the largest index such that s ¡ r and `pxr, xsq ¥ n0. Note that since `pxr, xr 1q ¥ n ¥ n0, such s exists.

If s m, then `pxr, xs 1q n0, and

`pxr, xs 1q ¥ mint`pxr, xsq, `pxs, xs 1qu¡∆ ¥ mintn0, `pxs, xs 1qu¡∆ n0¡∆. 80 1. Dynamical systems

Deﬁne then yi 1 xs 1. We have

n0 ¡ k ¤ `pyi, yi 1q n0.

If s 1 m, then we stop and get our sequence y0, . . . , yt, for t i 1.

If s m, then `pxr, xmq `pyi, xmq ¥ n0, and

`pyi¡1, xmq ¥ mint`pyi¡1, yiq, `pyi, xmqu¡∆ ¥ mintn0 ¡∆, n0u¡k n0 ¡2∆ and

`pyi¡1, xmq n0, since yi was deﬁned and was not equal to xm. Then we redeﬁne yi xm and stop the algorithm. In all the other cases we repeat the procedure. It is easy to see that at the end we get a sequence yi satisfying the conditions of the lemma.

Let x0 x, x1, . . . , xm y be an arbitrary sequence of points of X. Let n0 be the minimal value of `pxi, xi 1q. Let y0 x, y1, . . . , yt y be a sub-sequence of the sequence xi satisfying conditions of Lemma 1.4.16. Suppose at ﬁrst that 2α∆ ¡ ln C n `px, yq 0 . 0 α ¡ β

Remember that n0 `pxi, xi 1q for some i, hence ¢ ¸m ¡ p q ¡ βp2α∆ ¡ ln C0q ¡ p q e β` xi¡1,xi ¥ e βn0 ¡ exp ¡β`px, yq ¡ Ce β` x,y , α ¡ β i1 and the statement is proved. ¡ ¥ p q 2αk ln C0 Suppose now that n0 ` x, y α¡β , which is equivalent to

(1.8) pα ¡ βqn0 ¡ pα ¡ βq`px, yq ¡ 2α∆ ln C0 ¥ 0.

If t 1, then n0 ¡ 2∆ ¤ `px, yq n0, hence 2α∆ ¡ 2β∆ 2α∆ ¡ ln C n ¤ `px, yq 2∆ `px, yq `px, yq 0 , 0 α ¡ β α ¡ β since ln C0 0 2β∆. But this contradicts our assumption. Therefore, we have t ¡ 1, so that the inductive assumption implies

¸m t¸¡1 ¡ p q ¡ p q ¡ e β` xi¡1,xi ¥ Ce β` yi,yi 1 ¡ tCe βn0 . i1 i0 1.4. Hyperbolic dynamics 81

p ¡ ¡ p qq ¥ p q ¥ α n0 2∆ ` x,y We have t dn0¡2∆ x, y C0e , hence ¸m ¡β`pxi¡1,xiq ¡βn0 αn0¡2α∆¡α`px,yq e ¥ C0Ce i1 C exp pln C0 ¡ βn0 αn0 ¡ 2α∆ ¡ α`px, yqq ¡β`px,yq C exp p¡β`px, yq pα ¡ βqn0 ¡ pα ¡ βq`px, yq ¡ 2α∆ ln C0q ¥ Ce , by (1.8). Which ﬁnishes the proof.

1.4.4. Metrics associated with expanding maps. Theorem 1.4.17. Let X be a compact space. A continuous map f ü X is expanding (in the sense of Deﬁnition 1.4.1) if and only if f is metrically expanding for some metric on X .

Proof. Suppose that there exists a metric d and numbers ¡ 0 and L ¡ 1 such that dpfpxq, fpyqq ¡ Ldpx, yq for all px, yq P X 2 such that dpx, yq ¤ . Then the set tpx, yq : dpx, yq ¤ u is an expansion entourage and the action of N is expansive. Suppose now that the action of N is expansive. Then there exists a symmetric expansion entourage U. Suppose that d is a metric associated with the log-scale deﬁned by U, see Deﬁnition 1.4.10. Let α be the exponent of the metric d, and let C ¡ 1 be such that C¡1e¡α`px,yq ¤ dpx, yq ¤ Ce¡α`px,yq for all x, y P X . Let k be a positive integer, and suppose that `px, yq ¥ k. Then `pf kpxq, f kpyqq `px, yq ¡ k, and dpf kpxq, f kpyqq ¤ C2e¡αk. dpx, yq ln C2 p kp q kp qq ¤ It follows that for any integer k greater than α we have d f x , f y 2 ¡αk ¡1 ¡αk Ldpx, yq, where L C e 1, for all px, yq P Uk. If C e , then `px, yq ¥ k for all x, y P X such that dpx, yq ¤ . It follows that f k is metrically expanding. Lemma 1.4.4 shows that f is also expanding.

Let us investigate how canonical is the metric constructed in the proof of Theorem 1.4.17. Proposition 1.4.18. Let U and V be expansion entourages for a map f ü X . Then the sets of lower exponents for U and V coincide. If dU and dV are metrics associated with U and V of exponent α, then there exists C ¡ 1 such ¡1 that C dU px, yq ¤ dV px, yq ¤ CdU px, yq (i.e., the metrics are bi-Lipschitz equivalent). 82 1. Dynamical systems

Proof. Let Ù and `V be defined by U and V , respectively. By Lemma 1.4.8, there exists k such that Uk V , hence Un k Vn for all n P N. It follows that `V px, yq ¥ Ù px, yq k. The same arguments show that Ù px, yq ¥ `V px, yq k for some k, i.e., that |Ù px, yq ¡ `V px, yq| is uniformly bounded. The statements of the proposition easily follow from this fact.

Proposition 1.4.18 implies that for any expanding map f ü X the critical lower exponent αc is well deﬁned (i.e., does not depend on the choice of the expansion entourage), and for every β P p0, αq the corresponding metric of exponent β is uniquely deﬁned, up to a bi-Lipschitz equivalence. Note that if d is a metric of exponent β, then any metric bi-Lipschitz equivalent to d is also a metric of exponent β. The map f is metrically expanding with respect to some metric of exponent β for every β P p0, αcq. This class of metrics is studied in detail in the works of P. Ha¨ısinskyand K. Pilgrim (see [HP09] and references therein). The critical exponent αc is one of several numerical invariants of expanding dynamical systems...

1.4.5. Expansive actions of Z. Recall that a homeomorphism f ü X of a metric space is said to be expansive if there exists a closed neighborhood U X ¢ X of the diagonal such that f npUq is equal to the diagonal. nPZ Let f ü X be an expansive homeomorphism of a compact metric space, and let U be the corresponding expansion entourage as above. Deﬁne `px, yq as the maximal n ¥ 0 such that pf kpxq, f kpyqq P U for all ¡n k n. If such n does not exist, then we set `px, yq 0. It is obvious that `px, yq `py, xq and that `px, yq 8 if and only if x y. tp q P ¢ p q ¥ u Deﬁne, as in the expanding case, Un x, y X X : ` x, y n , i.e., kp q Un |k| n f U . Lemma 1.4.19. The map ` is a log-scale on X compatible with the topology (i.e., such that every metric associated with ` is compatible with the topology on X . The log-scale ` does not depend, up to bi-Lipschitz equivalence, on the choice of U.

Proof. It is proved in the same way as for expanding maps that for every neighborhood V of the diagonal there exists n such that Un V (see Lemma 1.4.8). The statement of the Lemma 1.4.9 is also true in our case with the same proof (the only thing to change is to replace “i 0, 1, . . . , n” by “¡n ¤ i ¤ n”. These two lemmas imply that the metric associated with the log-scale ` is compatible with the topology. Independence on the choice of U is also proven in the same way as for the expanding case, see the proof of Proposition 1.4.18. 1.4. Hyperbolic dynamics 83

We see that, in the same way as for expanding maps, we have a canonical class of metrics associated with an expansive homeomorphism f ü X . These metrics were deﬁned by D. Fried in [Fri83]. Deﬁnition 1.4.20. We say that x, y P X are stably equivalent if dpf npxq, f npxqq Ñ 0 as n Ñ 8. We say that they are unstably equivalent if dpf ¡npxq, f ¡npyqq Ñ 0 as n Ñ 8. Two points are homoclinic if they are simultaneously stably and unstably equivalent.

It is easy to see that the deﬁned relations are equivalences. Lemma 1.4.21. Let U be the expansivity entourage for an expansive homeomorphism f ü X . Two points x, y P X are stably (resp. unstably) equivalent n n if and only if there exists n0 P Z such that pf pxq, f pyqq P U for all n ¥ n0 (resp. all n ¤ n0).

Proof. The “only if” direction is obvious. Suppose that pf npxq, f npyqq P U n n for all n ¥ n0. Then `pf pxq, f pyqq ¥ n ¡ n0 for all n ¥ n0. It follows that ¡ p ¡ q for every metric associated with ` we have dpf npxq, f npyqq ¤ Ce α n n0 Ñ 0 as n Ñ 8. Define, for x P X , and an expansivity entourage U n n W ,U pxq ty P X : pf pxq, f pyqq P U @n ¥ ¡∆ ¡ 1u and n n W¡,U pxq ty P X : pf pxq, f pyqq P U @n ¤ ∆ 1u, where ∆ satisfies the condition of Definition 1.4.12 for the log-scale ` associated with U. Equivalently, W ,U pxq is the set of points y such that n n `pf pxq, f pyqq ¥ ∆ 1 for all n ¥ 0, and W¡,U pxq is the set of points y such that `pf npxq, f npyqq ¥ ∆ 1 for all n ¤ 0.

Lemma 1.4.22. For every pair x, y P X the intersection W ,U pxqXW¡,U pyq consists of at most one point.

Proof. Suppose that z1, z2 P W ,U pxq X W¡,U pyq. Then for every n P Z either n n n n minp`pf pz1q, f pxqq, `pf pz2q, f pxqqq ¥ ∆ 1 or n n n n minp`pf pz1q, f pyqq, `pf pz2q, f pyqqq ¥ ∆ 1, n n depending on the sign of n. But this implies that `pf pz1q, f pz2qq ¥ 1 n n for all n P Z, i.e., that pf pz1q, f pz2qq P U for all n P Z. It follows that z1 z2.

Deﬁnition 1.4.23. We will denote by rx, ysU or just rx, ys the unique intersection point of W ,U pxq and W¡,U pyq, if it exists. 84 1. Dynamical systems

1.4.6. Local product structures. 1.4.6.1. Rectangles. A rectangle is a topological space R together with a decomposition into a direct product R A ¢ B of two topological spaces. In order to make this structure more intrinsic (so that we do not introduce new spaces A and B), we can define the direct product structure as a binary operation rpa1, b1q, pa2, b2qs pa1, b2q on A ¢ B R. Then the structure of the direct product decomposition can be axiomatized in the following way. Definition 1.4.24. A rectangle is a topological space R together with a continuous map r¤, ¤s : R ¢ R ÝÑ R such that (1) rx, xs x for all x P R; (2) rx, ry, zss rx, zs and rrx, ys, zs rx, zs for all x, y, z P R. Example 1.4.25. Let f ü X be an expansive homeomorphism of a compact space. We say that R X is a rectangle for f if the map r¤, ¤s, given in Definition 1.4.23 is defined on whole R ¢ R. Then, by Proposition ??, pR, r¤, ¤sq is a rectangle in the sense of Definition 1.4.24.

Suppose that pR, r¤, ¤sq is a rectangle. Then plaques of x P R are deﬁned as

P1pR, xq ty P R : rx, ys xu, P2pR, xq ty P R : rx, ys yu. Note that we have the implication rx, ys x ùñ ry, xs ry, rx, yss ry, ys y. It follows that rx, ys x is equivalent to ry, xs y. It is shown in the same way that rx, ys y is equivalent to ry, xs x. In other words, y P PipR, xq is equivalent to x P PipR, yq for every i 1, 2.

Lemma 1.4.26. For every x P R the map r¤, ¤s : P1pR, xq ¢ P2pR, xq ÝÑ R is a homeomorphism.

Proof. For every y P R we have rx, ry, xss x, hence ry, xs P P1pxq. Simi- larly, rrx, ys, xs x, hence rx, ys P P2pxq. Then rry, xs, rx, yss rry, xs, ys y. It follows that y ÞÑ pry, xs, rx, ysq is a continuous map R ÞÑ P1pR, xq ¢ p q r¤ ¤s P2 R, x inverse to the map , .

Note that if pa1, b1q, pa2, b2q P P1pR, xq¢P2pR, xq, then rra1, b1s, ra2, b2ss ra1, b2s, i.e., the map r¤, ¤s, after the identiﬁcation of R with P1pR, xq ¢ P2pR, xq, becomes rpa1, b1q, pa2, b2qs pa1, b2q. See Figure 1.26, where the structure of a rectangle is shown.

Note that two diﬀerent plaques PipR, xq and PipR, yq are naturally iden- tiﬁed by a canonical homeomorphism, so that the decomposition of R into 1.4. Hyperbolic dynamics 85

Figure 1.26. A rectangle the direct product of plaques does not depend on the reference point (x or y). Namely, the respective homeomorphisms are

H1,x,y : z ÞÑ rz, ys : P1pR, xq ÝÑ P1pR, yq and

H2,x,y : z ÞÑ ry, zs : P2pR, xq ÝÑ P2pR, yq.

It is checked directly that H1,x,y ¥ H1,y,x and H2,x,y ¥ H2,y,x are identity homeomorphisms, and that the decomposition of R into the direct product P1pR, xq ¢ P2pR, xq is transformed by the homeomorphisms Hi,x,y to the decomposition of R into the direct product of plaques of y.

We will therefore denote sometimes by P1pRq and P2pRq the plaques of R as abstract topological spaces, without any reference to points of R. 1.4.6.2. Local product structures. Deﬁnition 1.4.27. Let X be a topological space. An atlas of a local product structure on X is a cover of X by open subsets Ri, i P I, together with structures of rectangles pRi, r¤, ¤siq on each of them, such that for every pair i, j P I and every x P X there exists a neighborhood U of x such that ry, zsi ry, zsj for all y, z P Ri X Rj X U. Two atlases are compatible if their union is also an atlas. A local product structure on X is a compatibility class of atlases of local product structures on X .

Note that the condition of Deﬁnition 1.4.27 is void in the case when x does not belong to the intersection of the closures of Ri and Rj. 86 1. Dynamical systems

Figure 1.27. Local product structure

An open subset R X together with a rectangle structure r¤, ¤s on R is a rectangle of X if the union of an atlas of X with tpR, r¤, ¤squ is an atlas of X , i.e., if the structure of a rectangle on R is compatible with the local product structure on X .

Example 1.4.28. Let F : B ÝÑ X be a locally trivial bundle with ﬁber P . It means that for every point x P X there is a neighborhood U of x ¡1 and a homeomorphism φU : P ¢ U ÝÑ F pUq such that F pφU pp, yqq y for all y P U and p P P . Moreover, the maps φU naturally agree with each p q p q P other, i.e., if U1 and U2 intersect, then φU1 p, y φU2 p, y for all p P and y P U1 X U2.

It is easy to see that the set of the rectangles φU pP ¢ Uq deﬁnes a local product structure on B.

Deﬁnition 1.4.29. Let X1, X2 be two spaces with local product structures on them. We say that a continuous map f : X1 ÝÑ X2 preserves the local product structures if for every point x P X1 there exist rectangular neighborhoods pR1, r¤, ¤s1q and pR2, r¤, ¤s2q of x and fpxq, respectively, such that fpry, zs1q rfpyq, fpzqs2 for all y, z P R1.

Example 1.4.30. Consider a rectangle X A ¢ B, and let G be a group acting properly and freely on X by homeomorphisms preserving the local product structure on X . Then the quotient GzX by the action is naturally a space with a local product structure. We call such local product structures n splittable. For example, for any decomposition of R into a direct sum of subspaces, we get the corresponding local product structure on the torus n n R {Z . See, for example, the decomposition into the direct sum of the eigenspaces for the “Arnold’s Cat” map 1.1.5. 1.4. Hyperbolic dynamics 87

1.4.7. Ruelle-Smale systems. The following class of dynamical systems was introduced by D. Ruelle in [Rue78] as a generalization of basic sets of hyperbolic diffeomorphisms. Definition 1.4.31. A Ruelle-Smale system (also called Smale space) is a homeomorphism f ü X of a compact metric space with a local product structure satisfying the following conditions. (1) The map f preserves the local product structure. (2) There exists λ P p0, 1q and a cover of X by a finite number of rectangles pRi, r¤, ¤sq such that for any two points x, y belonging to one plaque P1pRi, xq P2pRi, yq we have dpfpxq, fpyqq ¤ λdpx, yq; and for any two points x, y belonging to one plaque P2pRi, xq ¡1 ¡1 P2pRi, yq we have dpf pxq, f pyqq ¤ λdpx, yq.

In other words, a homeomorphism is a Ruelle-Smale system if it is contracting in one direction and expanding in the other direction of a local product structure preserved by it.

We will denote P1 W and P2 W¡. The plaques W pR, xq and W¡pR, xq are called the stable and the unstable plaques, respectively.

Note that by the Lebesgue’s covering lemma, if tpRi, r¤, ¤siquiPI is a finite atlas of the local product structure of a compact space X , then there exists ¡ 0 such that rx, ysi rx, ysj for all i, j P I and all x, y such that dpx, yq and the corresponding expressions are defined. It follows that we may assume that we have one map r¤, ¤s defined on a neighborhood of the diagonal of X ¢ X . It follows from the definition that if x and y belong to the same stable plaque, then the distance dpf npxq, f npyqq exponentially converges to zero. Similarly, if x and y belong to the same unstable plaque, then dpf ¡npxq, f ¡npyqq exponentially converges to zero. Proposition 1.4.32. A homeomorphism f ü X of a compact space is a Ruelle-Smale system if and only if f is expansive and for every expansivity entourage U the map r¤, ¤sU from Definition 1.4.23 is defined on a neighborhood of the diagonal. In particular, the local product structure satisfying the conditions of Def- inition 1.4.31 is unique and depends only on the topological conjugacy class of f ü X .

Proof. Let us prove at ﬁrst that every Ruelle-Smale systems is expansive. Let be a Lebesgue number of a ﬁnite cover tRiuiPI of X by rectangles. Sup- n n pose that x, y P X are such that dpf pxq, f pyqq for all n P Z and x y. n n Then for every n there exists in P I such that f pxq, f pyq both belong 88 1. Dynamical systems

r np q np qs P to Rin . Consider then the corresponding point zn f x , f y Rin . n We have then fpznq zn 1 for all n P Z, i.e., zn f pz0q. The distance n n dpf pxq, f pz0qq is bounded from above by the maximum of diameters of the n n ¡1 n¡1 n¡1 rectangles Ri. But we must have dpf pxq, f pz0qq ¥ λ dpf pxq, f pz0qq for all n P Z, which is a contradiction. It follows that is an expansivity constant for f ü X . The fact that the local product structure on the Ruelle-Smale system coincides with the local product structure given in Deﬁnition 1.4.23 for expansive systems is now straightforward. In the other direction, suppose that f ü X is an expansive homeomorphism of a compact space such that r¤, ¤sU is deﬁned on a neighborhood of the diagonal for every expansivity entourage U. Note that if U1 U2, then r¤ ¤s r¤ ¤s p r sq P p r sq P , U1 is a restriction of , U2 . We have x, x, y U and y, x, y U, which implies that r¤, ¤sU is continuous. The axioms of a local product structure are checked directly using Lemma 1.4.22. Let d be a metric on X associated with U (see beginning of 1.4.5). It is checked directly that some iterate of f ü X is Ruelle-Smale system with respect to d. Using then the same trick as in Lemma 1.4.4, we can modify the metric so that f ü X is a Ruelle-Smale system.

1.4.8. Examples of Ruelle-Smale systems. 1.4.8.1. Shifts of ﬁnite type.

Proposition 1.4.33. A subshift S AZ is a Ruelle-Smale system if and only if it is of ﬁnite type.

p q p q Z Proof. Two points an nPZ and bn nPZ of X are stably equivalent if and only if there exists n0 P Z such that an bn for all n ¥ n0. Similarly, they are unstably equivalent if and only if there exists n0 such that an bn for all n ¤ n0.

Suppose that F XZ is a subshift. Denote by UN the entourage consisting of all pairs pw1, w2q P F ¢ F such that w1pnq w2pnq for all ¡N ¤ n ¤ N. Suppose that F is a Ruelle-Smale system. Then there exists N such p q P p q X that for any pair of sequences w1, w2 UN the intersection W ,UN w1 p q W¡,UN w2 is non-empty. In other words, there exists N such that if sequences w1 and w2 from F coincide on the interval r¡N,Ns, then the sequence w equal to w1 on r¡N, 8q and to w2 on p¡8,Ns also belongs to F. The converse statement is also true: if F satisﬁes the last condition, then it is a Ruelle-Smale system. We leave it to the reader to use this condition to prove that every shift of ﬁnite type is a Ruelle-Smale system. 1.4. Hyperbolic dynamics 89

Let L X2N 2 be the set of all subwords of length 2N 2 of elements of F. Let us prove that if w P XZ is a sequence such that every subword of length 2N 2 of w belongs to L, then w P F. This will prove that F is a shift of ﬁnite type. It is enough to show that if v is ﬁnite word such that every subword of v of length 2N 2 belongs to L, then v is a subword of an element of F. Let us prove this statement by induction on the length of v. The statement is trivially true for |v| 2N 2. Suppose that we have proved it of |v| k, let us prove it for vx, where x P X. Write v yu for y P X. Then |v| |ux| k, so there exist sequences w1, w2 P F such that the restriction of w1 and w2 onto an interval ra, bs Z containing r¡N,Ns is equal to u, w1pa ¡ 1q y and w2pb 1q x. Then there exists a sequence w P F such that w|p¡8,Ns w1|p¡8,Ns and w|r¡N,8q w2|r¡N,8q. Then | yux vx is equal to w ra¡1,b 1s.

1.4.8.2. Anosov diffeomorphisms. Definition 1.4.34. An Anosov diffeomorphism is a diffeomorphism f ü M of a compact Riemannian manifold such that there exists a decomposition T M T ` T¡ of the tangent bundle into a direct sum of f-invariant sub-bundles, and constants C ¡ 0 and λ P p0, 1q such that n n (1) }Df p~vq} ¤ Cλ }~v} for all n ¥ 0 and ~v P T , ¡n n (2) }Df p~vq} ¤ Cλ }~v} for all n ¥ 0 and ~v P T¡.

By classical theory (see, for example, [Sma67, Theorem 7.4], [BS02, Theorem 5.6.4, Theorem 5.7.2]) every Anosov diffeomorphism is a Ruelle- Smale system. An example of an Anosov diffeomorphism is the Arnold’s Cat Map from 1.1.5. It can be generalized in the following way. Let L be a simply connected nilpotent Lie group, and let φ : L ÝÑ L be its automorphism such that the differential Dφ at the identity 1 P L is hyperbolic (i.e., its spectrum is disjoint with the unit circle). Let G be a subgroup of the affine group L n Aut L acting naturally on L, and suppose that the action G y L is free and co-compact, so that GzL is a compact manifold. Such manifolds are called infra nil-manifolds. Assume also that the G-action is φ-invariant, so that φ induces a diffeomorphism of GzL. Then this diffeomorphism is Anosov, see [Sma67, pp. 760–764]. We call such Anosov diffeomorphisms algebraic. All currently known Ansov diffeomorphisms are topologically conjugate to algebraic diffeomorphisms. It is an open question if this is a complete description. Moreover, the only known examples of Ruelle-Smale systems f ü X such that X is a connected and locally connected space are algebraic Anosov diffeomorphisms. 90 1. Dynamical systems

1.4.8.3. Hyperbolic sets. More generally, let M be a Riemanian manifold, U M a non-empty open subset, and let f : U ÝÑ fpUq be a diﬀeo- morphism. A closed totally f-invariant subset X U (i.e., such that fpUq U f ¡1pUq) is said to be hyperbolic if there exist decompositions TxM T pxq ` T¡pxq of the tangent spaces at x P X and constants C ¡ 0, λ P p0, 1q such that

(1) DfT pxq T pfpxqq, DfT¡pxq T¡pfpxqq; n n (2) }Df ~v} ¤ Cλ }~v} for all ~v P T pxq, x P X , and n ¥ 0; ¡n n (3) }Df ~v} ¤ Cλ }~v} for all ~v P T¡pxq, x P X , and n ¥ 0. A hyperbolic set X is locally maximal if there exists an open neighborhood V X such that Λ f npV q. nPZ If X is a locally maximal compact hyperbolic set, then f ü X is a Ruelle- Smale system, see [BS02, Proposition 5.9.1, 5.9.3]. Examples of locally maximal hyperbolic sets are the Smale horseshoe attractor W from 1.1.3, 3 and the solenoid from 1.1.4 (in its concrete version in R ). 1.4.8.4. DA attractors. We have seen in 1.1.5 that there exists a semiconju- 2 2 gacy φ : F ÝÑ R {Z from a shift of finite¢ type F to the Anosov diffeomor- 2 1 phism A ü 2{ 2 defined by the matrix . We will see later that R Z 1 1 this is true for any Ruelle-Smale system. The shift of finite type F can be defined as a result of cutting the torus along the boundaries of the elements of the Markov partition, and propagating the cuts by the Z-action of the dynamical system, exactly in the same way as we did it with the circle rotation in 1.3.1.2. We could also cut, i.e., make slits in the torus, only along the stable boundaries of the Markov partition. This will produce another Ruelle-Smale 2 2 system f ü X with a semiconjugacy φ : X ÝÑ R {Z . It is called a DA- attractor (“Derived from Anosov”). We will give an alternative description of this dynamical system later in.... The DA-attractor can be realized as a hyperbolic set of a diffeomorphism by modifying the Anosov diffeomorphism in a small neighborhood of a point (essentially by imitating the slits described above), see [Sma67, p. 788].

1.4.9. Natural extension of an expanding covering map. Proposition 1.4.35. If f ü X is an expanding map on a compact space, then its natural extension fˆ ü Xˆ is expansive. For the notion of a natural extension, see Deﬁnition 1.1.8.

Proof. Let ¡ 0 be an expansivity constant for f ü X with respect to a metric d on X . Consider the entourage U Xˆ ¢ Xˆ consisting of pairs of 1.4. Hyperbolic dynamics 91

sequences ppx1, x2,...q, py1, y2,...qq such that dpx1, y1q . Suppose that n n ξ px1, x2,...q and ζ py1, y2,...q are such that pfˆ pξq, fˆ pζqq P U for all n n n P Z. Then for every i ¥ 1 we have dpf pxiq, f pyiqq for all n ¥ 0, since n i¡1 n n n i¡1 n n f pξq pf pxiq, f pxi 1q,...q and f pζq pf pyiq, f pyi 1q,...q. ü This implies, by expansivity of f X , that xi yi for all i. Theorem 1.4.36. Let f ü X be an expanding covering map of a compact space, and let fˆ ü Xˆ be its natural extension. Then Xˆ is a ﬁber bundle with respect to the natural projection P : Xˆ ÝÑ X : for every x P X there exists a neighborhood U of x such that P ¡1pUq is naturally homeomorphic to the direct product C ¢ U, where C is the inverse limit of the sets f ¡npxq. The natural extension fˆ ü Xˆ is a Ruelle-Smale system, where the local product structure coincides with the local product structure coming from the described ﬁber bundle.

Proof. Let ¡ 0, L ¡ 1 be such that dpfpxq, fpyqq ¥ Ldpx, yq for all x, y P X such that dpx, yq ¤ . For every x P X there exists an open neighborhood U of x that is evenly covered, i.e., such that f ¡1pUq can be decomposed into a disjoint union ¡1 f pUq U1 Y U2 Y ¤ ¤ ¤ Y Um such that f : Ui ÝÑ U is a homeomorphism for every i. The decomposition is finite, since X is compact (hence f ¡1pxq is compact for every x P X ). Note that in general (if X is not locally connected) the decomposition is not unique. But we can use the fact that f is expanding to choose canonical decompositions for sets U of small diameter as follows. Since X is compact, there exists a finite cover U of X by open evenly covered sets. Then, by Lebesgue’s lemma, there exists δ0 ¡ 0 such that for every set B of diameter less than δ0 there exists U P U such that B U. It follows that every set of diameter less than δ0 is evenly covered. Consider decompositions of f ¡1pUq, for U P U, into disjoint unions U U1 Y ¤ ¤ ¤ Y Um such that f : Ui ÝÑ U are homeomorphisms, and ¡1 consider the corresponding inverse maps f : U ÝÑ Ui. By the continuity ¡1 of the maps f : U ÝÑ Ui, there exists δ δ0 such that for every set A of diameter less than δ the set f ¡1pAq can be decomposed into a disjoint union of sets A1 Y ¤ ¤ ¤ Y Am of sets of diameter less than . Then the diameters ¡1 of Ai will be less than L δ. Note that then the distance between any two different points of f ¡1pxq for x P X is not less than . Consequently, for any ¡1 x1 P Ai and x2 P Aj for i j we have dpx1, x2q ¡ ¡ 2L δ. If δ is small enough, then ¡ 2L¡1δ ¡ δ, and we get the following.

Lemma 1.4.37. If δ ¡ 0 is small enough, then for every set A X of diameter less than δ the set f ¡1pAq is decomposed in a unique way into a 92 1. Dynamical systems

¡1 disjoint union f pAq A1 Y ¤ ¤ ¤ Y Am such that f : Ai ÝÑ A are homeomorphisms, the sets Ai have diameters less than δ, and distance between any two points belonging to diﬀerent sets Ai is greater than δ.

Deﬁnition 1.4.38. We say that δ is a strong injectivity constant of the expanding covering f ü X if it satisﬁes the conditions of Lemma 1.4.37.

¡1 We will call the sets Ai the components of f pAq. For n ¡ 1, the ¡n ¡1 components of f pAq are deﬁned inductively as components of f pAiq, ¡pn¡1q where Ai is a component of f pAq. Note that since components of f ¡1pAq are of diameter less than L¡1δ δ, we have a unique decomposition of f ¡npAq into components. If A is connected, then components of f ¡npAq are its connected components. Fix some strong injectivity constant δ ¡ 0. Let U X be a set of diameter less than δ. Consider the rooted tree TU with the set of vertices equal to the disjoint union of the sets of components of f ¡npUq for n ¥ 0, where a component A of f ¡npUq is connected to the component fpAq of f ¡pn¡1qpUq. The root is f ¡0pUq tUu.

Similarly, for every x P X , denote by Tx the tree Ttxu with the set of vertices equal to the disjoint union of the sets f ¡npxq for n ¥ 0. For every x P U the trees Tx and TU are naturally isomorphic: the isomorphism maps ¡n ¡n a vertex t P f pxq of Tx to the unique component of f pUq containing t.

The boundary BTU of the tree TU is the inverse limit of the sets of components of f ¡npUq with respect to the maps induced by f. Similarly, ¡n ¡n BTx is the inverse limit of the sets f pxq with respect to f : f pxq ÝÑ f ¡pn¡1qpxq. For every set U X of diameter less than δ we get a natural homeo- ¡1 morphism φU : P pUq Xˆ ÝÑ U ¢ BTU deﬁned in the following way. Let p ξ pt0, t1,...q P X be a point of the inverse limit Xˆ, where t0 P U U0. ¡n Let Un be the component of f pUq such that tn P Un. Then φU pξq pt0, pU0,U1,...qq. Recall that t0 P pξq. It is easy to see that this is a homeomorphism.

It also follows from the fact that TU is naturally isomorphic to Tx for every x P U, that the homeomorphisms φU agree on the intersections of the sets U, i.e., that we have a ﬁber bundle. p 1 1 q p 2 2 q P If ξ1 t0, t1,... , ξ2 t0, t1,... U are such that the second coordi- p q p q 1 2 nates of φU ξ1 and φU ξ2 are equal, then ti and ti belong for every i to ¡np q p 1 2q Ñ Ñ 8 the same component of f U . This implies that d ti, ti 0 as i , hence ξ1 and ξ2 are unstably equivalent. 1.4. Hyperbolic dynamics 93

p 1 1 q p 2 2 q p q Suppose that ξ1 t0, t1,... and ξ2 t0, t1,... are such that P ξ1 n n P pξ2q P U. Consider the points fˆ pξ1q and fˆ pξ2q. They are equal to p np q n¡1p q p q 1 1 q f t0 , f t0 , . . . , f t0 , t0, t1, t2,... and p np q n¡1p q p q 1 1 q f t0 , f t0 , . . . , f t0 , t0, t1, t2,... , 1 2 ˆnp q respectively, where t0 t0 t0. We see that the distance between f ξ1 n and fˆ pξ2q in the inverse limit Xˆ goes to zero, i.e., that ξ1 and ξ2 are stably equivalent. We have shown that the local product structure defined by the homeomorphisms φU agrees with the stable and unstable equivalence classes, which finishes the proof of the theorem. Example 1.4.39. Let f ü X be an expanding endomorphism of a Rie- mannian manifold. Then its natural extension fˆ ü Xˆ can be realized as an attractor of a diffeomorphism, see [Sma67, p. 788]. We have seen an example of such a realization in the case of the angle doubling map and the solenoid in 1.1.4. The general case is very similar to the solenoid example.

1.4.10. Williams solenoids. The natural extension of a map f ü X is a Ruelle-Smale system not only in the case of expanding coverings. As a starting point of a more general setting, let us consider the following class of examples. Let σ : X ÝÑ X¦ be a substitution such that the length of σnpxq goes to inﬁnity for all x P X. Let Bσ be the corresponding stationary Vershik- Bratteli diagram, see 1.3.7. Suppose that Bσ is properly ordered, see Propo- sition 1.3.31. An example of such a substitution is (1.9) σp0q 01, σp1q 011, The associated Vershik-Bratteli diagram is shown on Figure 1.17. Let X be a bouquet of |X| oriented loops labeled by the letters of X. Consider the map fσ ü X realizing σ: it maps every loop labeled by x to the path in X on which the word σpxq is read. We parametrize each loop by r0, 1s, so that the common point of the loops is parametrized by 0 and 1, and assume that the word σpxq is read in the positive (increasing) direction.

See, for instance Figure 1.28 where the corresponding map fσ for the substitution (1.9) is shown.

By choosing appropriate lengths of the loops in X , we can make fσ expanding on each loop. The map fσ will be expanding the length of paths, but it is not expanding on any neighborhood of the common point of the loops, since it is not injective there. 94 1. Dynamical systems

Figure 1.28. Geometric realization of a substitution

n ü n Note that for every positive n the map fσ X is the realization of σ : the loop labeled by x is mapped to the path on which σnpxq is read.

Proposition 1.4.40. The natural extension fˆσ ü Xˆ is a Ruelle-Smale system. The space Xˆ is homeomorphic to the mapping torus of the adic transformation deﬁned by the Vershik-Bratteli diagram Bσ.

Proof. Let us denote by γx the loop of X labeled by x P X. Consider a point pt1, t2,...q P Xˆ. Suppose that t1 belongs to the interior of γx (i.e., is not equal to the common point of the loops) for some x P X. Let an P X be such that tn belongs to γan . In particular, x a1. Then fσ maps γan to a path containing xn¡1. We get an occurence of the letter an¡1 in the p q word σ an corresponding to the segment of γan mapped by fσ to γan¡1 and containing tn¡1. Let en¡1 be the edge of the Vershik-Bratteli diagram Bσ corresponding to this occurence. It is an edge connecting an to an¡1 in Bσ. We get a path pe1, e2,...q in Bσ. It is clear that the point pt1, t2,...q P Xˆ is uniquely determined by t1 and pe1, e2,...q. We will denote the point pt1, t2,...q by pt; e1, e2,...q, where t P p0, 1q is the parameter corresponding to the point t1. We see that the subset of Xˆ consisting of points pt1, t2,...q such that t1 different from the common point of the loops γx is naturally identified with the product of the open unit interval p0, 1q with the space of paths in Bσ. It is easy to show that this identification is a homeomorphism.

Denote by p0; e1, e2,...q and p1; e1, e2,...q the limits of pt; e1, e2,...q as t Ñ 0 and t Ñ 1, respectively. It follows from the construction that p1; e1, e2,...q is equal to p0; f1, f2,...q, where pf1, f2,...q is the image of pe1, e2,...q under the adic transformation, provided pe1, e2,...q is not maximal. If pe1, e2,...q is maximal, then the point p1; e1, e2,...q is equal to 1.4. Hyperbolic dynamics 95 pp, p, . . .q P Xˆ where p is the common point of the loops of X . The same is true for p0; e1, e2,...q if pe1, e2,...q is minimal. We see that the space Xˆ can be obtained from the product of r0, 1s with the space of paths PpBσq by identifying p1; e1, e2,...q with p0; f1, f2,...q, where pf1, f2,...q is the image of pe1, e2,...q by the adic transformation. It is not hard to check that fˆσ expands the distances in the direction of the p q unit interval and contracts them in the direction of P Bσ .

The construction from Proposition 1.4.40 and its generalizations were studied by R. F. Williams in [Wil67] and [Wil74]. The main feature of this approaches is that even if the map f ü X is not an expanding covering in the sense of Definition 1.4.1, it is “eventually” an expanding covering map. R. F. Williams, for example, uses branched manifolds, i.e., spaces equal n to unions of closed pieces of R pasted together in such a way that every point still has one well defined tangent space. We will not give a precise definition, but note that the above example of a rose of loops in Proposi- tion 1.4.40 is a branced manifold (you should imagine it as a union of circles tangent to each other at the common point). Then the expansion condition can be defined in the same way as for manifolds (see Example 1.4.5). The covering condition is replaced by a “flattening” condition: for every x P X there exists k ¥ 1 and a neighborhood N of x such that f kpNq is contained n in a subset of X diffeomorphic to an open ball of R . Note that this condition is satisfied for the example shown on Figure 1.28 with k 1: the image of a neighborhood of the singular point (the common point of the two circles) under the map is a smooth interval equal to the union of one black and one red half-intervals; it is trivially true for the interior points of the loops. The following more combinatorial version of Williams’ conditions (in the one-dimensional case) are given in [Yi01]. Definition 1.4.41. Let Γ be a graph seen as a topological space (a one- dimensional CW-complex) with a metric d compatible with the topology. Consider the following conditions. (1) Expansion: there exist constants C ¡ 0 and L ¡ 1 such that if x, y are points on an edge of Γ, and f n, for n ¥ 1 maps the interval rx, ys Γ to an edge of Γ, then dpf npxq, f npyqq ¥ CLndpx, yq. (2) Markov: The set V of vertices of Γ is forward-invariant: fpV q V . (3) Nonfolding: The map f n :ΓzV ÝÑ ΓzV is locally one-to-one for every n ¥ 1. 96 1. Dynamical systems

(4) Flattening: There exists k ¥ 1 such that for every x P Γ there exists an open neighborhood N of x such that f kpNq is homeomorphic to an open interval.

It is proved in [Yi01] that if f ü Γ satisfies the conditions of Defini- tion 1.4.41, then the natural extension fˆ ü Γˆ is a Ruelle-Smale system. (The paper includes additional irreducibility and non-wandering conditions.) More general conditions are given in S. Wieler’s paper [Wie14]. We present here a modified version. For an entourage U and x P X , we denote by BpU, xq the “ball of radius U” with center in x: BpU, xq ty P X : px, yq P Uu. Theorem 1.4.42. Let f ü X be a map, where X is compact and Hausdorff. Suppose that the following conditions hold: (1) Eventual expansion: There exists an entourage U and a number k ¥ 1 such that if pf npxq, f npyqq P U for all n ¥ 0, then f kpxq f kpyq. (2) Eventually open map: For every entourage U there exists an entourage V U and k ¥ 1 such that for every x P X we have f kpBpf kpxq,V qq f 2kpBpx, V qq. Then the natural extension fˆ ü Xˆ is a Ruelle-Smale system.

Proof. Let us prove at first that the first condition implies that the natural extension is expansive. The proof essentially repeats the proof of Proposition 1.4.35. Consider the entourage Uˆ Xˆ ¢ Xˆ consisting of all pairs ppx1, x2,...q, py1, y2,...qq such that px1, y1q P U. Suppose that ξ n n px1, x2,...q and ζ py1, y2,...q are such that pfˆ pξq, fˆ pζqq P Uˆ for all n n n P Z. Then we have pf pxiq, f pyiqq P U for all n ¥ 0 and i ¥ 1. It follows k k from eventual expansion of f that f pxiq f pyiq, hence xi¡k yi¡k for all i ¥ k. The latter implies that ξ ζ, i.e., that fˆ is expansive. Let us prove now that the map r¤, ¤s from Definition 1.4.23 is defined on a neighborhood of the diagonal of Xˆ ¢ Xˆ, if the map f ü X satisfies both conditions of the theorem. Let U be an entourage satisfying the first condition of the theorem, and let V be an entourage satisfying the second condition and such that V U and fpV q U. We can replace f by f k, so it is enough to prove the statement for the case k 1.

Let ξ px1, x2,...q and ζ py1, y2,...q be points of Xˆ such that pxi, yiq P V for i 1 and i 2. 1 p 1q P Let us construct by induction a sequence yi such that xi, yi V 2p 1 q p 1q ¥ 1 1 and f yi 1 f yi for all i 1. Set y1 y1 and y2 y2. Then 1.4. Hyperbolic dynamics 97

1 P p q p p q q p 1q P p p p q qq since yi B xi,V B f xi 1 ,V , we have f yi f B f xi 1 ,V 2p p qq 1 P p q f B xi 1,V . It follows that we can choose yi 1 B xi 1,V such that 2p 1 q p 1q 2 p 1 q p 2 q 2p 1 q f yi 1 f yi . Denote yi f yi 1 . Then we have f yi 1 f yi 2 p 1 q 2 2 p 1 q p q p 2q p p p 1 qq P f yi 1 yi , y1 f y2 f y2 y1, and xi, yi f xi 1, f yi 1 fpV q U for all i ¥ 1. 2 p 2 2 q P ˆ 2 We have found a point ζ y1, y2,... X such that y1 y1 and p 2q P r s 2 r¤ ¤s xi, yi U for all i. We have ξ, ζ ζ (check the order....), hence , is defined on a neighborhood of the diagonal, i.e., fˆ ü Xˆ is a Ruelle-Smale system. S. Wieler proved in [Wie14] that for every Ruelle-Smale system g ü S with totally disconnected stable direction there exists a map f ü X on a compact metric space X satisfying the conditions of Theorem 1.4.42 such that fˆ ü Xˆ is conjugate to g ü S. 2 Example 1.4.43. Consider a map f ü X defined on a domain in R as it is schematically shown on the left-hand side of Figure 1.29. We can choose f in such a way that it is contracting along an f-invariant foliation consisting of vertical lines in the orange rectangle and radial lines in the semi-annular shapes. We also may assume that iterations of f are expanding in a transversal foliation, so that the intersection of the ranges of f n is a hyperbolic set in the sense of 1.4.8.3. This set is locally maximal, and is called the Plykin attractor [Ply74]. If we collapse the domain of f along the leaves of the stable foliation, i.e., collapse the orange rectangle to a horizontal segment, and the semianular regions to loops, we will get a graph shown on the top part of the right- hand side of Figure 1.29. Since the foliation is f-invariant, f induces a well-defined self-map of the graph, as it is shown schematically on the lower part of the right-hand side of the figure. This map satisfies the conditions of Definition 1.4.41, and its natural extension is topologically conjugate to the Plykin attractor. Example 1.4.44. DA attractor...

1.4.11. Symbolic encoding and shadowing. Let H y X1 and H y X2 be topological dynamical systems, where H is a semigroup. Recall that a semiconjugacy from the ﬁrst system to the second one is a continuous map φ : X1 ÝÑ X2 such that φphpxqq hpφpxqq for all x P X1 and h P H. If there exists a surjective semiconjugacy φ : X1 ÝÑ X2, then H y X2 is called a factor of H y X1. The kernel of the semiconjugacy is the set tp q P 2 p q p qu Eφ x, y X1 : φ x φ y . 98 1. Dynamical systems

Figure 1.29. Plykin attractor and its one-dimensional model

2 p q P It is a subset of X1 invariant under the diagonal action of H: if x, y Eφ, then phpxq, hpyqq P Eφ. We consider the kernel as the topological dynamical system H y Eφ. Definition 1.4.45. We say that a system H y X is finitely presented if H there exists a subshift of finite type H y S, S X , and a surjective semiconjugacy φ : S ÝÑ X such that the kernel H y Eφ is also a shift of finite type. Here we naturally identify XH ¢ XH with pX ¢ XqH , so that H Eφ pX ¢ Xq . The terminology is attributed to M. Gromov... references... H Proposition 1.4.46. Let H y S X be a subshift, and let φ : S ÝÑ X be a surjective semiconjugacy to a dynamical system H y X . If H y X is expansive, then the kernel H y Eφ is a subshift of relative finite type in H y S ¢ S. For the notion of relative finite type, see Definition 1.2.13.

Proof. Let U X ¢ X be an expansion entourage. Consider its full preim- ¡1 ¡1 age φ pUq S ¢ S. We have Eφ φ pUq. Since Eφ is compact, and S ¢ S is zero-dimensional, there exists a clopen set U 1 S ¢ S such that 1.4. Hyperbolic dynamics 99

1 ¡1p q ¡1p 1q Eφ U φ U . Let us prove that Eφ hPH h U . We obviously ¡1p 1q 1 have Eφ hPH h U , since Eφ U and Eφ is H-invariant. Suppose ¡1 1 1 ¡1 that pw1, w2q P h pU q for all h. Then phpw1q, hpw2qq P U φ pUq for all h P H, hence phpφpw1qq, hpφpw2qqq P U for all h P H, which implies φpw1q φpw2q, i.e., pw1, w2q P Eφ. This proves that Eφ is of relative finite type in S ¢ S. Example 1.4.47. A subshift F XZ is called sofic if there exists a shift of finite type F˜ YZ and a surjective semi-conjugacy F˜ ÝÑ F. It follows from Proposition 1.4.46 that every sofic subshift is finitely presented. Consider, t uZ p q for example, the even subshift F 0, 1 consisting of all sequences xn nPZ such that there is an even number of 1s between any two consecutive 0s. It is nor hard to show that it is not a shift of finite type (exercise ...). On the other hand, if F˜ is the shift of finite type consisting of all sequences p q P t uZ xn nPZ 0, 1 with no subword 11, then the sliding block map defined by p00q ÞÑ 0, p01q ÞÑ 1, p10q ÞÑ 1 maps F˜ surjectively to F. Exercise... describe Eφ in this case...

Example 1.4.48. Let f ü R{Z be the angle doubling map x ÞÑ 2x pmod 1q. We have seen in 1.1.2 that it admits a surjective semiconjugacy ¦ t0, 1u ÝÑ R{Z from the one-sided full shift. The system f ü R{Z is expansive, therefore it follows from Proposition 1.4.46 that it is ﬁnitely presented. Check that the corresponding kernel is of ﬁnite type...

It follows from Proposition 1.4.46 that a system H y X is finitely presented if and only if it is expansive and is a factor of a shift of finite type. The following approach to factors of shifts of finite type is due to R. Bowen, see... Definition 1.4.49. Let H be a semigroup generated by a finite set S H (our main examples will be H N and H Z with S t1u), and let H y X be an action on a metric space. We say that a sequence h ÞÑ xh : H ÝÑ X is an -pseudo-orbit if

dpspxhq, xshq for all h P H and s P S. We say that a system H y X satisﬁes the shadowing property if for every δ ¡ 0 there exists a positive number ¡ 0 such that for every - pseudo-orbit xh there exists a point y P X such that dpxh, hpyqq δ for all h P H.

Picture of a pseudo-orbit... Proposition 1.4.50. Every Ruelle-Smale system satisﬁes the orbit shadowing property. 100 1. Dynamical systems

Proof. Let xn, n P Z be an -pseudo-orbit for a Ruelle-Smale system f ü X , where will be selected later. We assume that the metric d on X belongs to the canonical class of metrics ... Let λ P p0, 1q be as in the deﬁnition ... Let δ be such that the δ-neighborhood every point of X is contained in a rectangle. We assume that δ. Choose for every n a rectangle Rn containing the δ-neighborhood of xn.

Let us choose a metric dn on Rn equal to the direct product metric of the expanding and the contracting directions:

dnpx, yq maxtdprxn, xs, rxn, ysq, dprx, xns, ry, xnsqu.

This metric is compatible with the topology on Rn (in fact, it is bi-Lipschitz equivalent to d). ¥ Deﬁne xn,0 xn and then inductively, for k 0, r p qs xn,k 1 xn,k, f xn¡1,k .

Then all xn,k belong to the stable plaque of xn, and we have p q ¤ p q d xn,k, xn,k 1 λd xn¡1,k¡1, xn¡1,k , ¡ p q ¤ see Figure... It follows that there exists C 0 such that d xn,k, xn,k 1 Cδλk for all k for some C depending only on the metric d. It follows that the sequence xn,k, k 1, 2,... converges, provided all xn,k are defined. Each C point xn,k and the limit are on the distance not more than 1¡λ from xn. It follows that if is small enough, the points xn,k are defined and converge in Rn. Let xn be its limit. Note that it follows from the definitions that r p qs xn 1, f xn xn 1. ¡ ¡ Changing the direction, we will find a sequence xn,k satisfying xn,k 1 r ¡1p ¡ q ¡ s r ¡1p ¡ q s ¡ f xn 1,k , yn,k f xn 1,k , xn and converging to a point xn on dis- C r ¡1p ¡q ¡ s ¡ tance not more than 1¡λ from xn. We will also have f xn , xn¡1 xn¡1. r ¡s p q ¤ 2C Then yn xn , xn is an orbit such that d xn, yn 1¡λ for all n. Proposition 1.4.51. Suppose that H y X is an expansive dynamical system on a compact metric space satisfying the shadowing property. Then H y X is a factor of a shift of finite type, and hence is finitely presented. Corollary 1.4.52. Every Ruelle-Smale system is finitely presented.

Proof. Let δ ¡ 0 be a number less than half of the expansivity constant ... Let ¡ 0 be the corresponding constant from Definition 1.4.49. Let N X be a finite -net. Consider the set S N H consisting of all -pseudo-orbits, i.e., all sequences w : H ÝÑ N such that dpspwphqq, wpshqq for every h P H, s P S. It is a topological Markov shift, see... In particular, it is a shift of finite type. For every w P S, by Definition 1.4.49, there exists y P X 1.4. Hyperbolic dynamics 101 such that dpwphq, hpyqq δ. Define φpwq y. Note that if y1 is another point satisfying this condition, then dphpyq, hpy1qq 2δ for all h P H, hence y y1, by expansivity. We get a well defined map φ : S ÝÑ X such that φphpwqq hpφpwqq for all w P S and h P H. The map φ is surjective, since for every y P X and h P H we can choose wphq such that dpwphq, hpyqq . It is continuous, since if w1, w2 P S are such that w1phq w2phq for all h P A for a set A H, then dphpφpw1qq, hpφpw2qqq 2 for all h P A. By increasing A, we can get an arbitrarily small upper estimate of dpφpw1q, φpw2qq, by Lemma 1.2.7. It follows that φ : S ÝÑ X is a surjective semiconjugacy. Example 1.4.53. We have seen that the binary solenoid from 1.1.4 has a finite presentation given by the semiconjugacy from the two-sided shift t0, 1uZ. The corresponding kernel consists of the diagonal, the pair p... 111 ...,... 000 ...q, and all the pairs of the form p. . . xn¡1xn0111 . . . , . . . xn¡1xn1000 ...q. It is easy to check that this set is a subshift of finite type.

Suppose that φ : F ÝÑ X is a surjective semi-conjugacy from a subshift of finite type s ü F XZ to a Ruelle-Smale system f ü X . Consider tp q P u P the cylindrical subsets Cx xn nPZ F : x0 x for x X. If the corresponding subsets φpCxq have disjoint interiors, then we say that tφpCxqu is a Markov partition of f ü X . Note that the subshift F is uniquely determined by the Markov parition. Namely, for every generic P np q t X the point f t belongs to the interior of a unique element Cxn of the Markov partition for every n P Z (by Bair’s Category Theorem). The p q pp q q sequence xn nPZ is the itinerary of t, and we obviously have φ xn nPZ t. Then F is equal to the closure of the set of such itineraries. The subshift F can be seen as a result of “cutting” X along the boundaries of the elements of the Markov partition, and then propagating the cuts by the dynamics, similarly to what we did with an irrational rotation in 1.3.1.2 (compare it with the definition of Xθ in Proposition 1.2.37). It was proven by R. Bowen that every Ruelle-Smale system has a Markov partition, see [Bow70] (he proved it for hyperbolic sets of diffeomorphisms, but the proof of the general statement is the same). The class of finitely presented actions of Z is wider than the class of Ruelle-Smale systems. The following theorem of D. Fried [Fri87] clarifies the relation between these two classes. Theorem 1.4.54. An expansive system f ü X is finitely presented if and only if X can be covered by a finite number of closed rectangles.

Here a rectangle is a subset R X such that the operation r¤, ¤s given in Deﬁnition 1.4.23 is deﬁned and continuous on R ¢ R and takes values in R. 102 1. Dynamical systems

2 2 Example 1.4.55.¢ Consider the Arnold’s Cat map f ü R {Z defined by 2 1 the matrix . Note that fp¡xq ¡x, so f induces a well defined 1 1 homeomorphism of the space obtained from the torus by identifying every 2 2 element x P R {Z with ¡x. This space is homeomorphic to the sphere, and can be visualized as a result of folding the rectangle r0, 1{2s ¢ r¡1{2, 1{2s into a square pillow, see Figure..., where the foliation by the stable and unstable equivalence classes (manifolds) are shown... Show the partition into a finite number of rectangles... This is an example of a pseudo-Anosov diffeomorphism, see... Example 1.4.56. Consider the even subshift F from Example 1.4.47. It p q P t uZ is the set of all sequences xn nPZ 0, 1 such that there is an even number of 1s between any two consecutive 0s. Let R0 and R1 be the sets p q of sequences xn nPZ such that the number of leading 1s of x0x1x2 ... is even and odd, respectively (where infinity is considered to be even and p q p q P r s odd). If w1 xn nPZ, w2 yn nPZ Ri, then the sequence w1, w2 . . . y¡2y¡1 . x0x1x2 ... belongs to Ri F. It follows that F is a union of two closed rectangles R0 and R1. Their intersection is the set of all sequences p q P xn nPZ F such that x0x1x2 ... 111 .... 1.4.12. Structural stability. Corollary 1.4.52 implies that Ruelle-Smale systems can be described, up to topological conjugacy, by a finite amount of information: by a shift of finite type F and a shift of finite type E F¢F.A shift of finite type is, by definition, described by a finite number of prohibited subwords. In particular, there only countably many Ruelle-Smale systems, up to topological conjugacy. The same is true for expanding self-coverings (write more above)... In other words, hyperbolic dynamical systems are essentially combinatorial objects. One of aspects of the combinatorial nature of hyperbolic dynamical systems is their rigidity, or structural stability. It can be formu- lated, for example in the following way. Theorem 1.4.57. Let f ü X be a Ruelle-Smale system, and let d be a metric on X . Then there exists ¡ 0 such that if f 1 ü X is another Ruelle- Smale system such that dpfpxq, f 1pxqq for all x P X , then f ü X and f 1 ü X are topologically conjugate.

Proof. (Sketch.) If dpfpxq, f 1pxqq for all x P X , then the f-orbit p np qq 1 p 1p np qq n 1p qq f x nPZ is an -pseudo-orbit for f , since we have d f f x , f x dpf 1pf npxqq, fpf npxqqq . By Proposition 1.4.50, for every δ ¡ 0 we can find ¡ 0 such that if f and f 1 satisfy the condition of the theorem for , then for every x P X there exists x1 P X such that dppf 1qnpx1q, f npxqq δ for 1 1 all n P Z. If δ is smaller than the expansivity constant for f , the point x is 1.5. Holomorphic dynamics 103 unique. Consider the map φ : x ÞÑ x1. It follows from the definition that φ is a semiconjugacy from f ü X to f 1 ü X . On the other hand, x and x1 play the same role in the definition, so it follows from the uniqueness that φ is a bijection conjugating the systems. It remains to show that φ is continuous. This can be deduced from the proof of Proposition 1.4.50, where the orbit r ¡s shadowing a pseudo-orbit was constructed as yn xn , xn , where xn and ¡ xn were constructed as limits of sequences defined using the map and the operation r¤, ¤s. Using uniform continuity of f, f 1, and r¤, ¤s, one can show that φpxq depends continuously on x. Structural stability suggests that one should be able to describe and study hyperbolic dynamical systems using some algebraic or combinatorial techniques. Finite presentations and symbolic dynamics is in some sense such technique, though it is not easy to work with it, and to deduce topological properties of a dynamical system from its symbolic presentation. We will see later in Chapter 4 that there exists an algebraic approach to expanding covering maps, which have computationally efficient encoding by a self-similar group (or a biset). A similar encoding of Ruelle-Smale systems is still missing...

1.5. Holomorphic dynamics

Here we present a very short collection of classical introductory results in holomorphic dynamics, which will be used later. For a more detailed exposition, see the books [Mil06, Bea91]...

1.5.1. Preliminaries from complex analysis. Theorem 1.5.1 (Uniformization Theorem). Any simply connected Rie- mann surface (i.e., a one dimensional smooth complex manifold) is con- formally isomorphic to exactly one of the following surfaces. p (1) The Riemann sphere C. (2) The (Euclidean) plane C. (3) the open unit disc D tz P C : |z| 1u, or, equivalently, the upper half plane H tz P C : =z ¡ 0u. Theorem 1.5.2 (Schwarz Lemma). If f ü D is holomorphic and fp0q 0, then |f 1p0q| ¤ 1. If |f 1p0q| 1, then f is a rotation z ÞÑ cz about 0 (for |c| 1). If |f 1pzq| 1, then |fpzq| |z| for all z 0.

As a corollary we get

Theorem 1.5.3 (Liouville Theorem). If f ü C is holomorphic and bounded, then it is constant. 104 1. Dynamical systems

The automorphism groups (i.e., groups of bi-holomorphic automorphisms) of the simply connected Riemannian surfaces are as follows: ppq ÞÑ az b (1) Aut C is the group of§ all Möbiustransformations§ z cz d for § a b § a, b, c, d P such that § § 0. It is isomorphic to PSLp2, q. C § c d § C (2) AutpCq is the group of all affine transformations z ÞÑ az b for a, b P C, a 0. p q ÞÑ az b P (3) Aut H is the group§ of all§ transformations z cz d , where a, b, c, d § a b § are such that § § ¡ 0. It is isomorphic to PSLp2, q. Ev- R § c d § R ÞÑ iθ z¡a P ery automorphism of D is of the form z e 1¡az , where θ R, |a| 1. If S is a connected Riemann surface, then its universal covering Sr is p one of the simply connected surfaces C, C, D, and the fundamental group r π1pSq acts on S by conformal automorphisms. We say that S is Euclidean r or hyperbolic, if S is isomorphic to C or D, respectively. r Note that the action of π1pSq on S is fixed point free. Since every non- identical Möbiustransformation has a fixed point, the only surface with p p universal covering C is the sphere C itself. Any transformation z ÞÑ az b for a 1 has a fixed point, hence in the Euclidean case the fundamental group acts on the universal covering C by translations. It is easy to see that this implies that a Euclidean surface is isomorphic either to the cylinder C{Z, or to a torus C{Λ, where Λ is the subgroup of the additive group of C generated by two non-zero complex numbers a, b such that a{b R R. All the other Riemann surfaces are hyperbolic. It is a direct corollary of the Liouville theorem that every holomorphic map from C to a hyperbolic surface is constant (since we can lift it to the universal covering). In particular, every holomorphic map f ü C such that C r fpCq has more than one point is constant (Picard’s Theorem). Theorem 1.5.4 (Poincarémetric). There exists a unique Riemannian metric (up to multiplication by a constant) on D invariant under every conformal 2|dz| automorphism of D. It is given by ds 1¡|z|2 . Every orientation preserving isometry of D is a conformal automorphism. Every hyperbolic surface S has then a unique Poincarémetric coming r from the Poincarémetric on the universal covering S D of S (since the r fundamental group π1pSq acts on S by conformal automorphisms). The following is a corollary of Schwarz Lemma. 1.5. Holomorphic dynamics 105

Theorem 1.5.5 (Pick Theorem). Let f : S ÝÑ S1 be a holomorphic map between hyperbolic surfaces. Then exactly one of the following cases is taking place. (1) f is a conformal isomorphism and an isometry with respect to the Poincar´emetrics. (2) f is a covering map and is a local isometry. (3) f is strictly contracting, i.e., for every compact set K S there is a constant cK 1 such that dpfpxq, fpyqq ¤ ckdpx, yq for all x, y P K.

1.5.2. Fatou and Julia sets. Deﬁnition 1.5.6 (Compact-open topology). Let X be a locally compact space, and let Y an arbitrary topological space. Compact open topology on the space MappX , Yq of continuous maps X ÝÑ Y is given by the basis of neighborhoods of a map f : X ÝÑ Y consisting of sets

NK,pfq tg P MappX , Yq : dpfpxq, gpxqq for all x P Ku where K X is compact and ¡ 0.

In fact, the compact-open topology does not depend on the metric on Y. Convergence in the compact-open topology is called the uniform convergence on compact subsets. Deﬁnition 1.5.7. A set F of holomorphic functions from a Riemann surface S to a compact Rieman surface T is called a normal family if its closure is compact in MappS, T q. In the case when T is not compact, we replace T by its one-point compactiﬁcation.

Thus, a family F HolpS, T q is normal if every sequence fn of elements of F has either a subsequence fnk convergent uniformly on compact subsets, or a subsequence fnk converging to inﬁnity uniformly on compact subsets (i.e., such that for all compact K1 S and K2 S the intersection p q X fnk K1 K2 is empty for all k big enough). p Deﬁnition 1.5.8. Let f ü C be a rational function. The Fatou set of f is p the set of points z P C such that there exists a neighborhood U of z such ¥n p that f : U ÝÑ C, for n ¥ 0, is normal. The complement of the Fatou set is called the Julia set. n Example 1.5.9. Consider the function fpzq z for n ¥ 2. If |z0| 1, n then f pzq uniformly converges on a neighborhood of z0 to the constant 0 n function, hence z0 belongs to the Fatou set. If |z0| ¡ 1, then f pzq uniformly converges to 8 on a neighborhood of z0, hence z0 also belongs to the Fatou set in this case. On the other hand, if |z0| 1, then for any neighborhood U 106 1. Dynamical systems

n p of z0 and any sequence nk Ñ 8 the sequence f k : U ÝÑ C does not have a continuous limit. Consequently, the Julia set of zn is the unit circle.

It easily follows from the definitions that the Julia set J is totally invariant, i.e., fpJq J f ¡1pJq. The Julia sets of f and f n coincide. It is always non-empty (unless f is a Möbiustransformation) and compact. On the other hand, the Fatou set can be empty. Another possible definition of the Julia set is given by the following theorem (see, for instance [Mil06, Theorem 14.1]. Theorem 1.5.10. Let fpzq be a rational function of degree ¡ 1. Then the Julia set of f is equal to the closure of the union of its repelling cycles.

Here a cycle z0 fpzn¡1q, z1 fpz0§q, z2 fpz1q, . . . , zn¡1 fpzn¡2q n § df pzq § 1p q 1p q ¤ ¤ ¤ 1p q is called repelling if the multiplier dz f z0 f z1 f zn¡1 is zzi greater than one in absolute value. It is called attracting if the multiplier is less than one in absolute value. A Fatou component of a rational function fpzq is a connected component of the Fatou set of f. If U is a Fatou component, then fpUq is also a Fatou component. By D. Sullivan’s Nonwandering Theorem... the forward orbit of every Fatou component is finite, i.e., eventually belongs to a cycle. It follows that every Fatou component is a branched covering of a Fatou component fixed under some iteration of f. The fixed Fatou components are classified in the following way. Theorem 1.5.11. Let U be a Fatou of f such that fpUq U. Then one of the following four cases takes place. (1) U is the immediate basin of attraction of an attracting fixed point. (2) U is one petal of a parabolic fixed point of multiplier 1. (3) U is a Siegel disc. (4) U is a Herman ring.

Here the immediate basin of attraction of a ﬁxed point z0 is the con- p n nected component containing z0 of the set of points z P C such that limnÑ8 f pzq 1 z0. Similarly, if z0 is a ﬁxed point such that f pz0q 1, then there is a Fatou component, called a petal of points whose forward orbits converge to z0.A Siegel disc is an open domain such that the action of f on it is biholomor- phically conjugate to an irrational rotation of a disc. Similarly, a Herman ring is domain the action of f on which is conjugate to an irrational rotation of an annulus. For more detail, see [Mil06], in particular Section 16.

1.5.3. Hyperbolic rational functions. 1.5. Holomorphic dynamics 107

Deﬁnition 1.5.12. A rational function f is hyperbolic if it is expanding on a neighborhood of its Julia set.

Hyperbolic rational functions are important examples of expanding dynamical systems in the sense of Deﬁnitions 1.4.1 and 1.4.2. A post-critical set of f is the set of all points of the form f npcq, where c is a critical point of f, and n ¥ 1.

Theorem 1.5.13. Let f be a rational function of degree ¥ 2. Then the following conditions are equivalent. (1) f is hyperbolic. (2) The closure of the post-critical set of f is disjoint from its Julia set. (3) The orbit of every critical point converges to an attracting cycle.

Sketch of the proof. It is easy to see that (3) implies (2), since basins of attraction belong to the Fatou set. Let us show only that (2) implies (1) in p the case when P has more than two points. Then X CrP is a hyperbolic surface containing the Julia set. Note that fpP q P , hence f ¡1pX q X . The map f : f ¡1pX q ÝÑ X is a covering. Consider the Poincarémetrics on X and f ¡1pX q. The map f is a local isometry with respect to these metrics. The inclusion map Id : f ¡1pX q ÝÑ X is not a covering map, hence it is strictly contracting, see Theorem 1.5.5. It follows that if we consider the restriction of the Poincarémetric of X onto the subset f ¡1pX q, then the map f : f ¡1pX q ÝÑ X is expanding. Since the Julia set is compact and contained in f ¡1pX q, the map f will be uniformly expanding on the Julia set. If closure of the post-critical set has less than three points, then they p belong to attracting cycles, and we can take X equal to C minus a small neighborhood of P , and repeat the proof. Let us show that (1) implies (2). Let W be a neighborhood of the Julia set J such that f is expanding on W . Taking an -neighborhood of J in W , we get an open neighborhood U of J such that f is expanding on U, and f ¡1pUq U. Then f ¡npUq U for all n ¥ 1. The set U does not contain critical points of f, since otherwise f is not one-to-one, hence not expanding on any neighborhood of a critical point. If c is critical, and f npcq P U, then c P f ¡npUq U, which is a contradiction. Consequently, U does not contain any post-critical points. This implies that intersection of U with the closure of the post-critical set is empty. The fact that (2) implies (3) follows from classification of components of the Fatou set, see Theorem 1.5.11. 108 1. Dynamical systems

Figure 1.30. Julia set of a hyperbolic rational function

If f is a hypebolic rational function, then all its Fatou components belong to the basins of attraction to attracting cycle. In other words, only the ﬁrst type in the classiﬁcation of Fatou components in Theorem 1.5.11 is possible in this case. See examples of Julia sets of hyperbolic rational functions on Figure 1.30 and Figure 1.31.

1.5.4. Subhyperbolic rational functions. Post-critically ﬁnite rational functions in general, orbifolds and orbifold metrics... (deﬁne using the universal cover and lengths of paths)...

1.5.5. Quadratic polynomials and the Mandelbrot set. Let fpzq be a complex polynomial. Its critical points are zeros of the derivative and the infinity. The infinity is totally invariant, i.e., fp8q f ¡1p8q 8 and superattracting. It follows that the basin of attraction of infinity is 1.5. Holomorphic dynamics 109

Figure 1.31. A Sierpinski carpet Julia set connected. Points belonging to it escape to infinity. The filled Julia set of the polynomial is the complement of the basin of attraction of infinity. In other words, it is the set of points whose forward orbit is bounded. The filled Julia set is compact. The Julia set is its boundary..... Every quadratic polynomial is conjugate (by an affine transformation) to a polynomial of the form z2 c. It has only one finite critical point 0. Theorem 1.5.14. The Julia set of z2 c is connected if and only if the orbit of the critical point 0 is bounded. Otherwise it is homeomorphic to the Cantor set.

Proof. If the orbit of 0 is not bounded, then c belongs to the basin of infinity. Consider a curve γ connecting c to infinity inside the basin of infinity. Then f ¡1pγq disconnects the complex plane in two connected components, both of which are homeomorphically mapped onto the complement of γ. Since the Julia set of f is totally invariant, and does not intersect γ, it follows ¡1 that f pγq disconnects the Julia set J into two closed subsets J0 and J1 such that f : J0 ÝÑ J and f : J1 ÝÑ J are homeomorphisms.... 110 1. Dynamical systems

Figure 1.32. The Mandelbrot set

The main ideas of the proof... Deﬁnition 1.5.15. The Mandelbrot set M, see Figure 1.32, is the set of points c such that the Julia set of z2 c is connected. In other words, it is the set of points c such that the orbit of 0 under the iterations of z2 c is bounded.

It is known, see... that the Mandelbrot set is connected, and that its complement in the complex plane is homeomorphic to the complement of the closed unit disc. External angles... rational angles, hyperbolic components, their parametrization by the multiplier of the attracting cycle, Misiurewicz points, external angles in the dynamical plane...

1.5.6. Lyubich-Minski lamination. The natural extension, leaves, conformal type of the leaves... Exercises 111

Figure 1.33. Julia set of z2 ¡ 1

Figure 1.34. Julia set of z2 i

Figure 1.35. “Airplane” and “Rabbit”

Exercises

1.1. Prove that an action of a group G on a space X is minimal if and only if the space of orbits GzX has trivial (i.e., antidiscrete) topology. p q 1.2. Let an nPZ be the sequence deﬁned in 1.1.1. Let dn be the number of p q dn letters d in the sequence a1, a2, . . . , an . Find limnÑ8 n . 112 1. Dynamical systems

1.3. Find a number x P R{Z such that the closure of its orbit under the angle doubling map is homeomorphic to the Cantor set. 1.4. Show that for the one-sided shift s ü Xω there is no nonempty proper closed subset F Xω such that s¡1pF q F . 1.5. Tent map and a semiconjugacy with the one-sided shift... Describe the identifications... 1.6. Prove that the natural extension of the one-sided shift s ü Xω is topologically conjugate to the two-sided shift ü XZ. 1.7. Prove Proposition 1.1.11. 1.8. Prove that a point of the torus is represented by at most three sequences 2 2 in the encoding of the map A ü R {Z constructed in Subsection 1.1.5 for the Markov partition given in Figure 1.11. 1.9. Prove Proposition 1.2.3. 1.10. Prove that t0, 1uZ has uncountably many subshifts. 1.11. Prove that for every subshift of finite type F XZ the union of finite orbits is dense. ü P 1.12. Let s F be a subshift of finite type. Let pn be the number of° points w np q 8 n F such that s w w. Prove that the formal power series n0 pnx is a rational function. 1.13. Prove that the intersection of two shift of finite type is a shift of finite type.

1.14. Let x0x1x2 ... 01101 ... be the Thue-Morse word from Example 1.2.21. Prove that xn is equal to the sum modulo 2 of the digits of the binary expansion of n.

1.15. Let x0x1x2 ... be, as in the previous problem, the Thue-Morse sequence. Let A and B be the sets of numbers i 0, 1, 2,... such that xi 0 and xi 1, respectively. Prove that for every k 1, 2,..., the sets X t k 1 ¡ u X t k 1 ¡ u Ak A° 0, 1, 2,...,° 2 1 and Bk B 0, 1, 2,... 2 1 satisfy xd xd for all d 0, 1, 2, . . . , k. (solve...) xPAk xPBk 1.16. Prove that the Thue-Morse word is cube-free, i.e., that it has no subwords of the form vvv for a non-empty v P t0, 1u¦. 1.17. Show that the Fibonacci substitutional shift is palindromic... 1.18. Prove that every Sturmian subshift is palindromic... 1.19. Let A be a dense countable subset of r0, 1s disjoint from t0, 1u. Replace every point a P A by two copies a ¡ 0 and a 0 with the natural order on the obtained set X . Prove that X is homeomorphic to the Cantor set with respect to the order topology. Exercises 113

1.20. Consider the substitution

σ : a ÞÑ aca, b ÞÑ d, c ÞÑ b, d ÞÑ c

from [Lys85]. Show that it generates a minimal subshift. Find a primitive substitution generating a conjugate subshift. 1.21. Prove that the subshift from the previous problem is Toeplitz. 1.22. Paper folding sequence... Prove that it is Toeplitz. 1.23. Let σ : X¦ ÝÑ X¦ be a substitution such that for every x P X the word σpxq is non-empty (such substitutions are called non-erasing). Let βS be a block code (see ...), and let Fσ be the subshift defined by the substitution σ. Prove that βSpFσq is a substitutional subshift. 1.24. Let W be a non-empty set of words. Show that W is the set of all finite subwords of elements of a subshift F XZ if and only if the following conditions are satisfied. (1) Every subword of an element of W belongs to W . (2) For every v P W there exists a word v1vv2 P W , where ¦ v1, v2 P X are non-empty. Formulate a similar criterion for one-sided subshifts. 1.25. We say that a dynamical system Z y X is essentially minimal if there exists a unique closed Z-invariant set Y X such that Z y Y is minimal. Prove that a dynamical system Z y X , where X is totally disconnected compact and metrizable, is essentially minimal if and only if there exists a properly ordered Vershik-Bratteli diagram B such that Z y X is topologically conjugate to the system generated by the adic transformation on PpBq. In other words, prove Theorem 1.3.20 for essentially minimal systems without the condition that B is simple. 1.26. Let G y R{Z be an action of a group on the circle by homeomorphisms. Prove that either G y R{Z has a finite orbit, or G y R{Z is essentially minimal. Give an example of an action Z y R{Z that has no finite orbits but is not minimal. 1.27. Prove the statement of Example 1.3.9, i.e., construct and explicit isomorphism of the direct limit with the group of continuous maps PpBq ÝÑ G. 1.28. Find a properly ordered Vershik-Bratteli diagram realizing the Lysenok subshift from Propblem 20. 1.29. Suppose that B is a properly ordered Vershik-Bratteli diagram such that the sizes of the sets of vertices Vi and the sets of edges Ei are uniformly bounded, and the adic transformation generates an expansive Z-action. Show then that the complexity pF pnq of the adic transformation (see Definition 1.2.28) is bounded from above by a linear function. (Hint: generalize the proof of Theorem 1.2.34.) 114 1. Dynamical systems

1.30. Prove the converse S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dyn. Systems 16 (1996) 663–682.... 1.31. Complexity of Toeplitz subshifts... (realizability of a particular class of functions)... n m n m 1.32. Consider the sets of words W0 t0 10 : n, m ¥ 0u, W1 t1 01 : n m n m n, m ¥ 0u, W2 t0 1 : n, m ¥ 0u and W3 t1 0 : n, m ¥ 0u. Prove that if X is a countable subshift of complexity pX pnq n 1, then there exists a ﬁnite sequence k1, k2, . . . , kn and i P t0, 1, 2, 3u such ¤ ¤ ¤ p Y t uq that the WX ψk1 1ψk2 1 ψkn 1 Wi ∅ , where ψki 1 are as in Theorem 1.3.36. (Such subshifts are called skew-Sturmian, see...). 1.33. Show that Toeplitz subshifts can have arbitrarily large repetitivity functions...

1.34. Prove that the critical exponent αc (see Theorem 1.4.17) is infinite for every one-sided subshift. 1.35. Find the critical exponent of the circle doubling map x ÞÑ 2x pmod 1q. 1.36. Prove that every map satisfying the conditions of Definition 1.4.41 satisfies the conditions of Theorem 1.4.42. 1.37. Consider the following endomorphism σ of the free group a ÞÑ b, b ÞÑ b¡1cb, c ÞÑ cac¡1. Consider a rose X of three circles labeled by a, b, c, and a map f ü X realizing the substitution σ similarly to Proposition 1.4.40, i.e., mapping the circle labeled by x to the path σpxq in X . Choose a realization such that iterations of f expand the lengths of paths in X . Show that the natural extension fˆ ü Xˆ is topologically conjugate to the Plykin attractor (see Exampe 1.4.43). 2 1.38. Find the set of values of c P C such that z c has an attracting fixed point (i.e., a cycle of length 1). 2 1.39. Find the set of values of c P C such that z c has an attracting cycle of length 2. 1.40. Show that every repelling cycle belongs to the Julia set.

1.41. Consider the Tchebyshev polynomials Tdpxq cospd arccos xq. Describe the Julia sets of Td for d ¥ 1. 1.42. Let C{Zris be the torus, and let A ü C{Zris be the map given by Apzq p1 iqz. Find the Julia set of A. Using the fact that any holomorphic map f ü C{Λ on a torus is induced by a linear map on C, describe all possible Julia sets of holomorphic maps on the torus. 1.43. Consider the group G of all maps of the form z ÞÑ p¡1qkz a ib, where k P t0, 1u, and a, b P Z. Show that C{G is homeomorphic to a Exercises 115

sphere. Consider the map Apzq p1 iqz. Show that it induces a well defined map on the sphere C{G. Since the group G and the map A act by holomorphic maps, there is a well defined structure of a complex manifold on C{G, and A induces a holomorphic map on C{G, hence is can be realized by a rational function. What is the Julia set of this rational function? p q c 1.44. Prove that fc z 1 z2 has a unique cycle of length 2 consisting of the roots of the polynomial x2 ¡ cx c (except for the case c 4 when it degenerates to a fixed point). Show that this cycle is attracting if and only if |c| ¡ 4. ÞÑ c 1.45. Prove that the set of values c such that z 1 z2 has an attracting fixed point is equal to the image of the open unit disc under the transformation ÞÑ 4u u p2¡uq3 . It is the largest “cardioid” on Figure 6.7. P p¡ ¡ { q p q c 1.46. Let c 1, 4 27 , denote fc x 1 x2 . a) Prove that for every positive real number x we have fcpxq x and ¥np q ¤ that there exists n such that fc x 0. b) Prove that for every n ¥ 1 there exists cn such that the sequence ¥k ak f p1q satisfies 1 ¡ a1 ¡ a2 ¡ ... ¡ an 0. c) Find the limit limnÑ8 cn. 1.47. Prove that every quadratic polynomial can be conjugated by an affine map to z2 c for some c.

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